Understanding Geometric Sequences and Series

in this video we're going to focus on geometric sequences and series so first let's discuss the difference between a geometric sequence and a geometric series what do you think the difference is here's an example of a geometric sequence the numbers 3 6 12 24 48 and so forth a geometric sequence is different from an arithmetic sequence such as this one here in that a geometric sequence has a common ratio versus a common difference if you take the second term and divide it by the first term six divided by three is two you're going to get the common ratio if you take the third term divided by the second term you'll get the same common ratio 12 divided by 6 is 2. so that's the defining mark of a geometric sequence in an arithmetic sequence there's a common difference if you take the second term and subtract it from the first term eight minus five is three if you take the third term subtracted by the second eleven minus eight is three so that's how you could distinguish an arithmetic sequence from a geometric sequence an arithmetic sequence has a common difference between terms a geometric sequence has a common ratio between terms within an arithmetic sequence you're dealing with addition and subtraction for a geometric sequence you're dealing with multiplication and division between terms so now that we know what a geometric sequence is and how to distinguish it from an arithmetic sequence what is the geometric series a geometric series is basically the sum of the numbers in a geometric sequence so 3 plus 6 plus 12 plus 24 and so forth would be a geometric series this is the first term this is the second term this is the third and so forth now the formula you need to calculate the f term of a geometric sequence or series it's the first term a sub 1 times the common ratio r raised to the n minus 1. so for instance let's just make a note that r is equal to 2. let's say we want to find the value of the fifth term we know the fifth term is 48 but let's go ahead and calculate it so you can see how this formula works so the first term is three r the common ratio is two and n is the subscript here we're looking for the fifth term so n is five so five minus one is four two to the fourth power if you multiply two four times two times two times two times two that's 16. 16 times 3 is 48 so that's the function of this formula it gives you the value of the f term so you can find the value of the eighth term the 20th term and so forth the next equation you need to be familiar with first let's get rid of this the next equation is the partial sum formula the partial sum of a geometric series is the first term times 1 minus r raised to the n over 1 minus r so let's say that we want to find the sum of the first five terms this is going to be 3 plus 6 plus 12 plus 24. plus 48 go ahead and plug that into a calculator so for the first five terms i got the partial sum as being 93. now let's confirm that with this equation so let's calculate s sub 5 the first term is 3 times 1 minus r r is 2 n is 5 divided by 1 minus r so that's 1 minus 2. 2 to the fifth power that's 32 1 minus 2 is negative 1. now 1 minus 32 is negative 31. 3 times negative 31 that's negative 93 but divided by negative 1 that becomes positive 93 so we get the same answer so anytime you need to find the sum of a finite series you could use this formula so this series here is finite we're looking for the sum of the first five terms there's beginning and there's an end this series here is not finite it's an infinite geometric series the reason being is because of the dot dot that we see here it goes on forever it doesn't stop at the fifth term it keeps on going to infinity so it's an infinite geometric series this is an infinite geometric sequence it's a sequence that goes on from forever and it's geometric so make sure you can identify if a sequence is arithmetic geometric is it finite infinite is it a sequence or a series now the next thing we need to talk about is the arithmetic mean and the geometric mean let's call the arithmetic mean m a the arithmetic mean is simply the average of two numbers the geometric mean let's call it mg is the square root of the product of two numbers so let's go back to the arithmetic sequence that we had here if we wanted to find the arithmetic mean between the first and the third term it will give us the middle number the second term if you average 5 and 11 and divide by 2 using this formula you're going to get 16 over 2 which is eight so thus when you find the arithmetic mean of the first term and the third term you're going to get the second term because the average of one and three is two now let's find the arithmetic mean between the first and the fifth term this will give us the middle term 11. so if we were to add up a1 and a5 and then divided by 2 if we were to get the average we would get a3 the average of 1 and 5 is three so let's add five and seventeen and then divide by two five plus seventeen is twenty two twenty two divided by two is eleven so that's the concept of the arithmetic mean whenever you take the arithmetic mean of two numbers within an arithmetic sequence you get the middle term of that of those two numbers that you selected now the same is true for a geometric sequence if we were to find the geometric mean between 3 and 12 we would get the middle number 6. if we wanted to find the geometric mean between 3 and 48 we would get the middle number 12. so let's confirm that let's find the geometric mean between a1 and a3 so the first term is 3 the third term is 12. 3 times 12 is 36 the square root of 36 is 6. so we get the middle number now let's find the geometric mean between the first term and the fifth term so we should get 12 as an answer so the average of one and five one plus five is six divided by two is three so we should get a sub three the first term is 3 the last term or the 5th term is 48. now what's 3 times 48 if you're not sure what you could do is break it up into smaller numbers 48 is three times sixteen three times three is nine so you have the square root of nine times the square root of sixteen the square root of nine is three the square root of sixteen is four three times four is twelve so the geometric mean of 3 and 48 is the middle number in the geometric sequence which is 12. now sometimes you need to be able to write equations between terms within a geometric sequence for instance if you want to relate the second equation to the first equation you need to multiply by r i mean the second term to the first term if you want to relate the fifth term to the second term you need to multiply by r cubed to go from the second term to the fifth term you need to multiply it by r three times if you multiply six by r you're going to get 12. if you multiply 12 by r you get 24. 24 by r you get 48. so to go from the second term to the fifth term you need to multiply by r cubed and the reason why it's cube is because the difference between five and two is three and you could check that so if you take the second term which is six multiply it by two to the third that's six times eight which is 48 and that gives you the fifth term so if i want to relate the ninth term to the fourth term how many r values do i got to multiply the fourth term to get to the ninth term nine minus four is five so i gotta multiply the fourth term by r to the fifth power to get the ninth term so make sure you know how to write those formulas so we've discussed calculating the sum of a finite series just review if you want to calculate the sum of a finite series one that has a beginning and an end you would use this formula now what about the sum of an infinite series how can we find that what's the formula that we need to calculate s to infinity it's basically this same formula but without that part it's a one over one minus r so here's two examples of an infinite geometric series this is one of them and this one is going to be another one eight four two one one half and so forth we can't calculate the sum of both infinite geometric series for this one r is two so r or rather the absolute value of r is greater than one when that happens the geometric series diverges which means you can't calculate the sum because it doesn't it doesn't converge to a specific value if you keep adding these numbers it's not going to converge to a value it's going to get bigger and bigger and bigger so the series diverges if you try to calculate it let's say you plugged in 1 for a1 and 2 for r it's not going to work you get 3 over negative 1 which is negative 3 and clearly that's not the sum of this series the fact that you get a negative sum from positive numbers tells you something is wrong so this formula doesn't work if the series diverges it only works if the series converges and that happens when the absolute value of r is less than one if we focus on this particular infinite geometric series notice the value of r if we take the second term divided by the first term four over eight is one half if we take the third term divided by the second term two over four reduces to one half so that's the value of r so for that particular series we could say that the absolute value of r which is one half that's less than one therefore the series converges which means we can calculate a sum it has a finite sum even though the numbers get smaller and smaller and smaller now let's calculate the sum so the sum of an infinite number of terms of this geometric series is going to be the first term a sub 1 which is 8 over 1 minus r where r is a half one minus one half is one half so multiplying the top and bottom by two we get 16 on top these two will cancel we get one so the sum of this infinite geometric series that converges is 16. so that's how you can calculate the sum of an infinite geometric series the series must converge and for that to happen the absolute value of r has to be less than one if it's greater than one the series will diverge and you won't be able to calculate the sum now let's work on some practice problems write the first five terms of each geometric sequence shown below so let's start with the first one the first term is two to find the next term we need to multiply the first term by the common ratio the second term is equal to the first term times the common ratio so 2 times 3 is 6. and then to get the third term we just got to multiply the second term by the common ratio 6 times 3 is 18 18 times 3 is 54 and then 54 times 3 that's 162. so that's the answer for number one let's move on to number two the first term is 80. the common ratio is one-half so we're going to multiply 80 by a half half of 80 is 40 half of 40 is 20 half of 20 is 10 half of 10 is five so those are the first five terms for the second geometric sequence now let's move on to number three so the first term is six to find the next term we need to multiply six by negative two so this is going to be negative twelve negative twelve times negative two is positive 24 and then it's just going to alternate so whenever you see a sequence a geometric sequence with alternating signs then you know that the common ratio must be negative number two write the first five terms of the geometric sequence defined by the recursive formula shown below so we're given the first term when n is 2 we have that the second term is equal to negative 4 times the first term and we know that the second term is the first term times r so therefore r the common ratio must be negative 4. so anytime you need to write a recursive formula of a geometric sequence it's going to be a sub n is equal to r times the previous term a sub n minus 1. the next term is always the previous term times the common ratio so the common ratio is this number negative four so once we have the first term in the common ratio we can easily write out the sequence so the first term is negative three the second term will be negative three times negative four which is twelve the third term will be net twelve times negative four which is negative forty eight the fourth term is negative 48 times negative 4 which is 192 and then the fifth term 192 times negative 4 is negative 768 so that's how we can write the first five terms of the geometric sequence defined by recursive formula it's by realizing that this number is the common ratio write a general formula that gives the f term of each geometric sequence and then calculate the value of the eighth term of each of those geometric sequences so let's start with number one so we have the number 6 24 96 384 and so forth the first thing we need to do is calculate the common ratio so let's divide the second term by the first term dividing 24 by six we get four now just to confirm that this is indeed a geometric sequence let's take the third term and divide it by this the second term not the first one so 96 divided by 24 and that is also equal to 4. so we have a geometric sequence here in order to write the formula all we need is the value of the first term and the common ratio so we could use this equation the f term is going to be equal to the first term times r raised to the n minus one the first term being six r is four so we can write it as a sub n is equal to 6 times 4 raised to the n minus 1. so this is the answer for part a for the first sequence now let's move on to part b let's calculate the value of the eighth term so we just got to plug in 8 into n so it's 6 times 4 raised to the eight minus one eight minus one is seven four raised to the seventh power is sixteen thousand three hundred eighty four times six this gives us ninety eight thousand three hundred and four so that is the value of the eighth term and you could confirm it if you keep multiplying these numbers by 4 you're going to get it 384 times 4 that's 15 36 that's the fifth term if you times it by 4 again you get sixty one forty four times four you get twenty four five seven six and then times four gives you this number now let's move on to number two so we have the sequence 5 negative 15 45 negative 135 and so forth so the first term is five the common ratio which can be calculated by taking the second term divided by the first term that's negative 15 divided by five that's negative three r is also equal to the third term divided by the second term so that's 45 over negative 15 which is negative three so the value of the first term is five r is negative three so now let's go ahead and write a general formula that gives us the nth term so a sub n is going to be a sub 1 times r raised to the n minus 1. the first term is 5 r is negative three so this right here is the answer for part a now part b calculate the value of the eighth term so let's replace n with eight negative three raised to the seventh power that's negative two thousand one hundred and eighty-seven multiplying that by five this gives us negative ten thousand nine hundred and thirty-five so that's the final answer for part b number four describe each pattern of numbers as arithmetic or geometric finite or infinite sequence or series let's look at the first one do we have a common difference or common ratio going from 4 to 8 we increase by four from eight to twelve that's an increase by four and twelve to sixteen so we're constantly adding four we're not multiplying by four so therefore we have a common difference and not a common ratio so because we have a common difference this is arithmetic not geometric we're dealing with addition rather than multiplication now is this a sequence or series we're not added numbers so we have a sequence if you see a comma between the numbers it's going to be a sequence if you see a plus you're dealing with a series now is this sequence finite or infinite it has a beginning and it has an end we don't have dots that indicate that it goes on forever so this is finite so we have a finite arithmetic sequence for number one now let's move on to number two going from 90 to 30 that's a difference of negative 60. going from 30 to 10 that's a difference of negative 20. so we don't have a common difference here if we divide the second term by the first term this reduces to one-third if we divide the third term by the second term that's also one-third so what we have here is a common ratio rather than a common difference so the pattern of numbers is geometric not arithmetic we're multiplying by 1 3 to get the second term from the first term now are we dealing with a sequence or series so we don't have a plus sign between a number so we have a comma so we're dealing with a sequence and this sequence has no end it goes on forever so what we have here is an infinite geometric sequence now for number three we could see that we have a common ratio of two five times two is ten ten times two is twenty twenty times two is forty so this is geometric now there's a plus between the numbers so this is going to be a series not a sequence and this sequence i mean the series rather comes to an end the last number is 80. so it's not infinite but it's finite so we have a finite geometric series for the last one we can see that we have a common difference of negative four fifty minus four is forty six forty six minus four is 42 so this is going to be arithmetic we have a plus between a number so it's a series and it goes on forever so it's infinite so we have an infinite arithmetic series number five find the sum of the first ten terms of the geometric sequence shown below so the first term is 7 the common ratio negative 14 divided by 7 that's negative 2. 28 divided by negative 14 is also negative 2. so now that we know the first term in the common ratio we can calculate the sum using this formula so it's the first term times one minus r raised to the n over one minus r so the sum of the first 10 terms it's going to be the first term 7 times 1 minus negative 2 now it's raised to the n power n is 10 and then divided by 1 minus r negative two raised to the tenth power that's positive one thousand twenty four and here we have one minus negative two which is one plus two and that's three one minus ten twenty four that's negative one thousand twenty three seven times negative ten twenty three divided by three that's negative two thousand three hundred and eighty seven so that's the final answer try this problem find the sum of the infinite geometric series so we have the numbers 270 90 30 10 and so forth so we can see that the first term a sub 1 that's 270. the common ratio if we divide 90 by 70 what does that simplify to i mean 90 by 270. well we could cancel a 0 so we get 9 over 27 9 is 3 times 3 27 is 9 times three so that becomes true over nine three we can write that as three times one nine is three times three this is one third so that's going to be the common ratio and of course if you divide 30 by 90 you also get one-third so now that we have the first term and the common ratio we can now calculate the sum of the infinite geometric series using this formula it's going to be a sub 1 over 1 minus r by the way this particular infinite geometric series does a converge converger diverge the absolute value of r is less than one it's one third which is about 0.333 or 0.3 repeating so because it's less than 1 the infinite geometric series converges we can calculate the sum the sum is finite so let's go ahead and calculate that sum the first term is 270 r is one third so what's one minus one third one if you multiply it by three over three you get three over three minus one over three which is two over three now what i'm going to do is i'm going to multiply the top and bottom by three these threes will cancel so it's going to be 270 times three divided by two two seventy divided by two is one thirty five one thirty five times three and that's going to be four oh five so that is the sum of this infinite geometric series you

in this video we&#39;re going to focus on geometric sequences and series so first let&#39;s discuss the difference between a geometric sequence and a geometric series what do you think the difference is here&#39;s an example of a geometric sequence the numbers 3 6 12 24 48 and so forth a geometric sequence is different from an arithmetic sequence such as this one here in that a geometric sequence has a common ratio versus a common difference if you take the second term and divide it by the first term six divided by three is two you&#39;re going to get the common ratio if you take the third term divided by the second term you&#39;ll get the same common ratio 12 divided by 6 is 2. so that&#39;s the defining mark of a geometric sequence in an arithmetic sequence there&#39;s a common difference if you take the second term and subtract it from the first term eight minus five is three if you take the third term subtracted by the second eleven minus eight is three so that&#39;s how you could distinguish an arithmetic sequence from a geometric sequence an arithmetic sequence has a common difference between terms a geometric sequence has a common ratio between terms within an arithmetic sequence you&#39;re dealing with addition and subtraction for a geometric sequence you&#39;re dealing with multiplication and division between terms so now that we know what a geometric sequence is and how to distinguish it from an arithmetic sequence what is the geometric series a geometric series is basically the sum of the numbers in a geometric sequence so 3 plus 6 plus 12 plus 24 and so forth would be a geometric series this is the first term this is the second term this is the third and so forth now the formula you need to calculate the f term of a geometric sequence or series it&#39;s the first term a sub 1 times the common ratio r raised to the n minus 1. so for instance let&#39;s just make a note that r is equal to 2. let&#39;s say we want to find the value of the fifth term we know the fifth term is 48 but let&#39;s go ahead and calculate it so you can see how this formula works so the first term is three r the common ratio is two and n is the subscript here we&#39;re looking for the fifth term so n is five so five minus one is four two to the fourth power if you multiply two four times two times two times two times two that&#39;s 16. 16 times 3 is 48 so that&#39;s the function of this formula it gives you the value of the f term so you can find the value of the eighth term the 20th term and so forth the next equation you need to be familiar with first let&#39;s get rid of this the next equation is the partial sum formula the partial sum of a geometric series is the first term times 1 minus r raised to the n over 1 minus r so let&#39;s say that we want to find the sum of the first five terms this is going to be 3 plus 6 plus 12 plus 24. plus 48 go ahead and plug that into a calculator so for the first five terms i got the partial sum as being 93. now let&#39;s confirm that with this equation so let&#39;s calculate s sub 5 the first term is 3 times 1 minus r r is 2 n is 5 divided by 1 minus r so that&#39;s 1 minus 2. 2 to the fifth power that&#39;s 32 1 minus 2 is negative 1. now 1 minus 32 is negative 31. 3 times negative 31 that&#39;s negative 93 but divided by negative 1 that becomes positive 93 so we get the same answer so anytime you need to find the sum of a finite series you could use this formula so this series here is finite we&#39;re looking for the sum of the first five terms there&#39;s beginning and there&#39;s an end this series here is not finite it&#39;s an infinite geometric series the reason being is because of the dot dot that we see here it goes on forever it doesn&#39;t stop at the fifth term it keeps on going to infinity so it&#39;s an infinite geometric series this is an infinite geometric sequence it&#39;s a sequence that goes on from forever and it&#39;s geometric so make sure you can identify if a sequence is arithmetic geometric is it finite infinite is it a sequence or a series now the next thing we need to talk about is the arithmetic mean and the geometric mean let&#39;s call the arithmetic mean m a the arithmetic mean is simply the average of two numbers the geometric mean let&#39;s call it mg is the square root of the product of two numbers so let&#39;s go back to the arithmetic sequence that we had here if we wanted to find the arithmetic mean between the first and the third term it will give us the middle number the second term if you average 5 and 11 and divide by 2 using this formula you&#39;re going to get 16 over 2 which is eight so thus when you find the arithmetic mean of the first term and the third term you&#39;re going to get the second term because the average of one and three is two now let&#39;s find the arithmetic mean between the first and the fifth term this will give us the middle term 11. so if we were to add up a1 and a5 and then divided by 2 if we were to get the average we would get a3 the average of 1 and 5 is three so let&#39;s add five and seventeen and then divide by two five plus seventeen is twenty two twenty two divided by two is eleven so that&#39;s the concept of the arithmetic mean whenever you take the arithmetic mean of two numbers within an arithmetic sequence you get the middle term of that of those two numbers that you selected now the same is true for a geometric sequence if we were to find the geometric mean between 3 and 12 we would get the middle number 6. if we wanted to find the geometric mean between 3 and 48 we would get the middle number 12. so let&#39;s confirm that let&#39;s find the geometric mean between a1 and a3 so the first term is 3 the third term is 12. 3 times 12 is 36 the square root of 36 is 6. so we get the middle number now let&#39;s find the geometric mean between the first term and the fifth term so we should get 12 as an answer so the average of one and five one plus five is six divided by two is three so we should get a sub three the first term is 3 the last term or the 5th term is 48. now what&#39;s 3 times 48 if you&#39;re not sure what you could do is break it up into smaller numbers 48 is three times sixteen three times three is nine so you have the square root of nine times the square root of sixteen the square root of nine is three the square root of sixteen is four three times four is twelve so the geometric mean of 3 and 48 is the middle number in the geometric sequence which is 12. now sometimes you need to be able to write equations between terms within a geometric sequence for instance if you want to relate the second equation to the first equation you need to multiply by r i mean the second term to the first term if you want to relate the fifth term to the second term you need to multiply by r cubed to go from the second term to the fifth term you need to multiply it by r three times if you multiply six by r you&#39;re going to get 12. if you multiply 12 by r you get 24. 24 by r you get 48. so to go from the second term to the fifth term you need to multiply by r cubed and the reason why it&#39;s cube is because the difference between five and two is three and you could check that so if you take the second term which is six multiply it by two to the third that&#39;s six times eight which is 48 and that gives you the fifth term so if i want to relate the ninth term to the fourth term how many r values do i got to multiply the fourth term to get to the ninth term nine minus four is five so i gotta multiply the fourth term by r to the fifth power to get the ninth term so make sure you know how to write those formulas so we&#39;ve discussed calculating the sum of a finite series just review if you want to calculate the sum of a finite series one that has a beginning and an end you would use this formula now what about the sum of an infinite series how can we find that what&#39;s the formula that we need to calculate s to infinity it&#39;s basically this same formula but without that part it&#39;s a one over one minus r so here&#39;s two examples of an infinite geometric series this is one of them and this one is going to be another one eight four two one one half and so forth we can&#39;t calculate the sum of both infinite geometric series for this one r is two so r or rather the absolute value of r is greater than one when that happens the geometric series diverges which means you can&#39;t calculate the sum because it doesn&#39;t it doesn&#39;t converge to a specific value if you keep adding these numbers it&#39;s not going to converge to a value it&#39;s going to get bigger and bigger and bigger so the series diverges if you try to calculate it let&#39;s say you plugged in 1 for a1 and 2 for r it&#39;s not going to work you get 3 over negative 1 which is negative 3 and clearly that&#39;s not the sum of this series the fact that you get a negative sum from positive numbers tells you something is wrong so this formula doesn&#39;t work if the series diverges it only works if the series converges and that happens when the absolute value of r is less than one if we focus on this particular infinite geometric series notice the value of r if we take the second term divided by the first term four over eight is one half if we take the third term divided by the second term two over four reduces to one half so that&#39;s the value of r so for that particular series we could say that the absolute value of r which is one half that&#39;s less than one therefore the series converges which means we can calculate a sum it has a finite sum even though the numbers get smaller and smaller and smaller now let&#39;s calculate the sum so the sum of an infinite number of terms of this geometric series is going to be the first term a sub 1 which is 8 over 1 minus r where r is a half one minus one half is one half so multiplying the top and bottom by two we get 16 on top these two will cancel we get one so the sum of this infinite geometric series that converges is 16. so that&#39;s how you can calculate the sum of an infinite geometric series the series must converge and for that to happen the absolute value of r has to be less than one if it&#39;s greater than one the series will diverge and you won&#39;t be able to calculate the sum now let&#39;s work on some practice problems write the first five terms of each geometric sequence shown below so let&#39;s start with the first one the first term is two to find the next term we need to multiply the first term by the common ratio the second term is equal to the first term times the common ratio so 2 times 3 is 6. and then to get the third term we just got to multiply the second term by the common ratio 6 times 3 is 18 18 times 3 is 54 and then 54 times 3 that&#39;s 162. so that&#39;s the answer for number one let&#39;s move on to number two the first term is 80. the common ratio is one-half so we&#39;re going to multiply 80 by a half half of 80 is 40 half of 40 is 20 half of 20 is 10 half of 10 is five so those are the first five terms for the second geometric sequence now let&#39;s move on to number three so the first term is six to find the next term we need to multiply six by negative two so this is going to be negative twelve negative twelve times negative two is positive 24 and then it&#39;s just going to alternate so whenever you see a sequence a geometric sequence with alternating signs then you know that the common ratio must be negative number two write the first five terms of the geometric sequence defined by the recursive formula shown below so we&#39;re given the first term when n is 2 we have that the second term is equal to negative 4 times the first term and we know that the second term is the first term times r so therefore r the common ratio must be negative 4. so anytime you need to write a recursive formula of a geometric sequence it&#39;s going to be a sub n is equal to r times the previous term a sub n minus 1. the next term is always the previous term times the common ratio so the common ratio is this number negative four so once we have the first term in the common ratio we can easily write out the sequence so the first term is negative three the second term will be negative three times negative four which is twelve the third term will be net twelve times negative four which is negative forty eight the fourth term is negative 48 times negative 4 which is 192 and then the fifth term 192 times negative 4 is negative 768 so that&#39;s how we can write the first five terms of the geometric sequence defined by recursive formula it&#39;s by realizing that this number is the common ratio write a general formula that gives the f term of each geometric sequence and then calculate the value of the eighth term of each of those geometric sequences so let&#39;s start with number one so we have the number 6 24 96 384 and so forth the first thing we need to do is calculate the common ratio so let&#39;s divide the second term by the first term dividing 24 by six we get four now just to confirm that this is indeed a geometric sequence let&#39;s take the third term and divide it by this the second term not the first one so 96 divided by 24 and that is also equal to 4. so we have a geometric sequence here in order to write the formula all we need is the value of the first term and the common ratio so we could use this equation the f term is going to be equal to the first term times r raised to the n minus one the first term being six r is four so we can write it as a sub n is equal to 6 times 4 raised to the n minus 1. so this is the answer for part a for the first sequence now let&#39;s move on to part b let&#39;s calculate the value of the eighth term so we just got to plug in 8 into n so it&#39;s 6 times 4 raised to the eight minus one eight minus one is seven four raised to the seventh power is sixteen thousand three hundred eighty four times six this gives us ninety eight thousand three hundred and four so that is the value of the eighth term and you could confirm it if you keep multiplying these numbers by 4 you&#39;re going to get it 384 times 4 that&#39;s 15 36 that&#39;s the fifth term if you times it by 4 again you get sixty one forty four times four you get twenty four five seven six and then times four gives you this number now let&#39;s move on to number two so we have the sequence 5 negative 15 45 negative 135 and so forth so the first term is five the common ratio which can be calculated by taking the second term divided by the first term that&#39;s negative 15 divided by five that&#39;s negative three r is also equal to the third term divided by the second term so that&#39;s 45 over negative 15 which is negative three so the value of the first term is five r is negative three so now let&#39;s go ahead and write a general formula that gives us the nth term so a sub n is going to be a sub 1 times r raised to the n minus 1. the first term is 5 r is negative three so this right here is the answer for part a now part b calculate the value of the eighth term so let&#39;s replace n with eight negative three raised to the seventh power that&#39;s negative two thousand one hundred and eighty-seven multiplying that by five this gives us negative ten thousand nine hundred and thirty-five so that&#39;s the final answer for part b number four describe each pattern of numbers as arithmetic or geometric finite or infinite sequence or series let&#39;s look at the first one do we have a common difference or common ratio going from 4 to 8 we increase by four from eight to twelve that&#39;s an increase by four and twelve to sixteen so we&#39;re constantly adding four we&#39;re not multiplying by four so therefore we have a common difference and not a common ratio so because we have a common difference this is arithmetic not geometric we&#39;re dealing with addition rather than multiplication now is this a sequence or series we&#39;re not added numbers so we have a sequence if you see a comma between the numbers it&#39;s going to be a sequence if you see a plus you&#39;re dealing with a series now is this sequence finite or infinite it has a beginning and it has an end we don&#39;t have dots that indicate that it goes on forever so this is finite so we have a finite arithmetic sequence for number one now let&#39;s move on to number two going from 90 to 30 that&#39;s a difference of negative 60. going from 30 to 10 that&#39;s a difference of negative 20. so we don&#39;t have a common difference here if we divide the second term by the first term this reduces to one-third if we divide the third term by the second term that&#39;s also one-third so what we have here is a common ratio rather than a common difference so the pattern of numbers is geometric not arithmetic we&#39;re multiplying by 1 3 to get the second term from the first term now are we dealing with a sequence or series so we don&#39;t have a plus sign between a number so we have a comma so we&#39;re dealing with a sequence and this sequence has no end it goes on forever so what we have here is an infinite geometric sequence now for number three we could see that we have a common ratio of two five times two is ten ten times two is twenty twenty times two is forty so this is geometric now there&#39;s a plus between the numbers so this is going to be a series not a sequence and this sequence i mean the series rather comes to an end the last number is 80. so it&#39;s not infinite but it&#39;s finite so we have a finite geometric series for the last one we can see that we have a common difference of negative four fifty minus four is forty six forty six minus four is 42 so this is going to be arithmetic we have a plus between a number so it&#39;s a series and it goes on forever so it&#39;s infinite so we have an infinite arithmetic series number five find the sum of the first ten terms of the geometric sequence shown below so the first term is 7 the common ratio negative 14 divided by 7 that&#39;s negative 2. 28 divided by negative 14 is also negative 2. so now that we know the first term in the common ratio we can calculate the sum using this formula so it&#39;s the first term times one minus r raised to the n over one minus r so the sum of the first 10 terms it&#39;s going to be the first term 7 times 1 minus negative 2 now it&#39;s raised to the n power n is 10 and then divided by 1 minus r negative two raised to the tenth power that&#39;s positive one thousand twenty four and here we have one minus negative two which is one plus two and that&#39;s three one minus ten twenty four that&#39;s negative one thousand twenty three seven times negative ten twenty three divided by three that&#39;s negative two thousand three hundred and eighty seven so that&#39;s the final answer try this problem find the sum of the infinite geometric series so we have the numbers 270 90 30 10 and so forth so we can see that the first term a sub 1 that&#39;s 270. the common ratio if we divide 90 by 70 what does that simplify to i mean 90 by 270. well we could cancel a 0 so we get 9 over 27 9 is 3 times 3 27 is 9 times three so that becomes true over nine three we can write that as three times one nine is three times three this is one third so that&#39;s going to be the common ratio and of course if you divide 30 by 90 you also get one-third so now that we have the first term and the common ratio we can now calculate the sum of the infinite geometric series using this formula it&#39;s going to be a sub 1 over 1 minus r by the way this particular infinite geometric series does a converge converger diverge the absolute value of r is less than one it&#39;s one third which is about 0.333 or 0.3 repeating so because it&#39;s less than 1 the infinite geometric series converges we can calculate the sum the sum is finite so let&#39;s go ahead and calculate that sum the first term is 270 r is one third so what&#39;s one minus one third one if you multiply it by three over three you get three over three minus one over three which is two over three now what i&#39;m going to do is i&#39;m going to multiply the top and bottom by three these threes will cancel so it&#39;s going to be 270 times three divided by two two seventy divided by two is one thirty five one thirty five times three and that&#39;s going to be four oh five so that is the sum of this infinite geometric series you

Transcript for:Understanding Geometric Sequences and Series

Transcript for:
Understanding Geometric Sequences and Series