in this video we're going to focus mostly on arithmetic sequences now to understand what an arithmetic sequence is it's helpful to distinguish it from a geometric sequence so here's an example of an arithmetic sequence the numbers 3 7 11 15 19 23 and 27 represents an arithmetic sequence this would be a geometric sequence 3 6 12 24 48 96 192. do you see the difference between these two sequences and do you see any patterns within them in the arithmetic sequence on the left notice that we have a common difference this is the first term this is the second term this is the third fourth and fifth term to go from the first term to the second term we need to add four to go from the second to the third term we need to add four and that is known as the common difference in a geometric sequence you don't have a common difference rather you have something that is called the common ratio to go from the first term to the second term you need to multiply by two to go from the second term to the third term you need to multiply by two again so that is the r value that is the common ratio so in an arithmetic sequence the pattern is based on addition and subtraction in a geometric sequence the pattern is based on multiplication and division now the next thing that we need to talk about is the mean how to calculate the arithmetic mean and the geometric mean the arithmetic mean is basically the average of two numbers it's a plus b divided by two so when taking an arithmetic mean of two numbers within an arithmetic sequence let's say if we were to take the mean of three and eleven we would get the middle number in that sequence in this case we would get 7. so if you were to add 3 plus 11 and divide by 2 3 plus 11 is 14 14 divided by 2 gives you 7. now let's say if we wanted to find the arithmetic mean between 7 and 23 it's going to give us the middle number of that sequence which is 15. so if you would add up 7 plus 23 divided by 2 7 plus 23 is 30 30 divided by 2 is 15. so that's how you can calculate the arithmetic mean and that's how you can identify it within an arithmetic sequence the geometric mean is the square root of a times b so let's say if we want to find the geometric mean between three and six it's going to give us the middle number of the sequence which is i mean if we were to find the geometric mean between 3 and 12 we will get the middle number of that sequence which is 6. so in this case a is 3 b is 12. 3 times 12 is 36 the square root of 36 is 6. now let's try another example let's find the geometric mean between 6 and 96 this should give us the middle number 24. now we need to simplify this radical 96 is six times sixteen six times six is thirty six the square root of thirty six is six the square root of sixteen is four so we have six times four which is twenty 24. so as you can see the geometric mean of two numbers within the geometric sequence will give us the middle number in between those two numbers in that sequence now let's clear away a few things the formula that we need to find the nth term of an arithmetic sequence is a sub n is equal to a sub 1 plus n minus 1 times the common difference d in a geometric sequence its a sub n is equal to a 1 times r raised to the n minus 1. now let's use that equation to get the fifth term in the arithmetic sequence so that's going to be a sub 5 a sub 1 is the first term which is three n is five since we're looking for the fifth term the common difference is four in this problem five minus one is four 4 times 4 is 16 3 plus 16 is 19. so this formula gives you any term in the sequence you could find the fifth term the seventh term the 100th term and so forth now in a geometric sequence we could use this formula so let's calculate the six term of the geometric sequence it's going to be a sub six which equals a sub one the first term is three the common ratio is two and this is going to be raised to the six minus one six minus one is five and then two to the fifth power if you multiply two five times two times two times two times two times two so we can write it out so this here that's four three twos make eight four times eight is thirty two so this is three times thirty two three times thirty is ninety three times two is six so this will give you ninety six so that's how you could find the f term in a geometric sequence by the way make sure you have a sheet of paper to write down these formulas so that when we work on some practice problems you know what to do now the next thing we need to do is be able to calculate the partial sum of a sequence s sub n is the partial sum of a series of a few terms and it's equal to the first term plus the last term divided by two times n for geometric sequence the partial sum s of n is going to be a sub 1 times 1 minus r raised to the n over 1 minus r so let's find the sum of the first seven terms in this sequence so that's going to be s sub 7 that's going to equal the first term plus the seventh term divided by 2 times n where n is the number of terms which is 7. now think about what this means so basically to find the sum of an arithmetic sequence you're basically taking the average of the first and the last term in that sequence and then multiplying it by the number of terms in that sequence because this is basically the average of 3 and 27. and we know the average or the arithmetic mean of 3 and 27 that's going to be the middle number 15. so let's go ahead and plug this in so this is 3 plus 27 over 2 times 7. 3 plus 27 is 30 plus 2 i mean well 30 divided by 2 that's 15. so the average of the first and last term is 15 times 7 10 times 7 is 70. 5 times 7 is 35 so this is going to be 105. that's the sum of the first seven terms and you can confirm this with your calculator if you add up three plus seven plus eleven plus fifteen plus 19 plus 23 and then plus 27 and that will give you s of 7 the sum of the first seven terms go ahead and add up those numbers if you do you'll get 105. so that's how you can confirm your answer now let's do the same thing with a geometric sequence so let's get the sum of the first six terms s sub six so this is going to be three plus six plus twelve plus twenty four plus 48 plus 96 so we're adding the first six terms now because it's not many terms we're adding we can just simply plug this into our calculator and we'll get 189 but now let's confirm this answer using the formula so s sub 6 the sum of the first six terms is equal to the first term a sub one which is three times one minus r r is the common ratio which is two raised to the n n is six over one minus r or 1 minus 2. so i'm going to work over here since there's more space now 2 to the 6 that's going to be 64. if you recall 2 to the fifth power was 32 if you multiply 32 by 2 you get 64. so this is going to be 1 minus 64. and 1 minus 2 is negative 1. so this is 3 times 1 minus 64. is negative 63. so we could cancel the two negative signs a negative divided by a negative will be a positive so this is just 3 times 63. 3 times 6 is 18. so 3 times 60 has to be 180 and then 3 times 3 is 9 180 plus 9 adds up to 189 so we get the same answer now what is the difference between a sequence and a series i'm sure you heard of these two terms before but what is the difference between them now we've already considered what an arithmetic sequence is a sequence is basically a list of numbers so that's a sequence a series is the sum of the numbers in a sequence so this here is an arithmetic sequence this is an arithmetic series because it's the sum of an arithmetic sequence now what we have here is a sequence but it's a geometric sequence as we've considered earlier this is a geometric series it's the sum of a geometric sequence now there are two types of sequences and two types of series you have a finite sequence and an infinite sequence and it's also a finite series and an infinite series this sequence is finite it has a beginning and it has an n this series is also finite it has a beginning and it has an end in contrast if i were to write 3 7 11 15 19 and then dot dot dot this would be an infinite sequence the presence of these dots tells us that the numbers keep on going to infinity now the same is true for a series let's say if i had three plus seven plus eleven plus fifteen plus nineteen and then plus dot dot dot dot that would also be an infinite series so now you know the difference between a finite series and an infinite series now let's work on some practice problems describe the pattern of numbers shown below is it a sequence or series is it finite or infinite is it arithmetic geometric or neither so let's focus on if it's a sequence or series first part a so we got the numbers 4 7 10 13 16 19. we're not adding the numbers we're simply making a list of it so this is a sequence the same is true for part b we're simply listing the numbers so that's a sequence in part c we're adding a list of numbers so since we have a sum this is going to be a series d is also a series e that's a sequence for f we're adding numbers so that's a series and the same is true for g so hopefully this example helps you to see the difference between a sequence and a series now let's move on to the next topic is it finite or is it infinite to answer that all we need to do is identify if we have a list of dots at the end or not here this ends at 19. so that's a finite sequence the dots here tells us it's going to go forever so this is an infinite sequence this one we have the dots so this is going to be an infinite series this ends at 162 so it's finite so we have a finite series this is going to be an infinite sequence next we have an infinite series and the last one is a finite series now let's determine if we're dealing with an arithmetic geometric or neither sequence or series so we're looking for a common difference or a common ratio so for a notice that we have a common difference of three four plus three is seven seven plus three is ten so because we have a common difference this is going to be an arithmetic sequence for b going from the first number to the second number we need to multiply by two four times two is eight eight times two is sixteen so we have a common ratio which makes this sequence geometric for answer choice c going from five to nine that's plus four and from nine to thirteen that's plus four so we have a common difference so this is going to be not an arithmetic sequence but an arithmetic series for answer choice d going from two to six we're multiplying by three and then six times three is eighteen so that's a geometric a geometric series now for e going from 50 to 46 that's a difference of negative 4 and 46 to 42 that's the difference of negative four so this is arithmetic for f we have a common ratio of four three times four is twelve twelve times 4 is 48 and if you're wondering how to calculate d and r to calculate d take the second term subtracted by the first term 7 minus 4 extreme or you could take the third term subtracted by the second 10 minus 7 is 4. in the case of f if you take 12 divided by 3 you get 4. 48 divided by 12 you get 4. so that's how you can calculate the common difference or the common ratio is by analyzing the second term with respect to the first one so since we have a common ratio this is going to be geometric for g if we subtract 18 by 12 we get a common difference of positive six 24 minus 18 gives us the same common difference of six so this is going to be arithmetic so now let's put it all together let's summarize the answers so for part a what we have is a finite arithmetic sequence part b this is an infinite geometric sequence c we have an infinite arithmetic series d is a finite geometric series e is an infinite arithmetic sequence f is an infinite geometric series g is a finite arithmetic series so we have three columns of information with two different possible choices thus two to the third is eight which means that we have eight different possible combinations right now i have seven out of the eight different combinations the last one is a finite geometric sequence which i don't have listed here so now you know how to identify whether you have a sequence or series if it's a rhythmical geometric and if it's finite or infinite number two write the first four terms of the sequence defined by the formula a sub n is equal to three n minus seven so the first thing we're going to do is find the first term so we're going to replace n with one so it's going to be three minus seven which is negative 4. and then we're going to repeat the process we're going to find the second term a sub 2. so it's 3 times 2 minus 7 which is negative 1. next we'll find a sub 3. three times three is nine minus seven that's two and then the fourth term a sub four that's going to be twelve minus seven which is five so we have a first term of negative four then it's negative one two five and then the sequence can continue so the common difference in this problem is positive three going from negative one to two if you add three you'll get two and then two plus three is five but this is the answer for the problem so this is those are the first four terms of the sequence number three write the next three terms of the following arithmetic sequence in order to find the next three terms we need to determine the common difference a simple way to find the common difference is to subtract the second term by the first term 22 minus 15 is 7. now just to confirm we need to make sure that the difference between the third and the second term is the same 29 minus 22 is also seven so we have a common difference of seven so we could use that to find in the next three terms so 36 plus 7 is 43 43 plus 7 is 50 50 plus 7 is 57 so these are the next three terms of the arithmetic sequence here's a similar problem but presented differently write the first five terms of an arithmetic sequence given a one and d so we know the first term is 29 and the common difference is negative four so this is all we need to write the first five terms if the common difference is negative four then the next term is going to be 29 plus negative four which is 25 25 plus negative four or 25 minus 4 is 21 21 minus 4 is 17 17 minus 4 is 13. so that's all we need to do in order to write the first five terms of the arithmetic sequence given this information number five write the first five terms of the sequence defined by the following recursive formulas so let's start with the first one part a so we're given the first term what are the other terms when dealing with recursive formulas we need to realize is that you get the next term by plugging in the previous term so let's say n is 2. when n is two this is a sub two and that's going to equal a sub n minus one two minus one is one so this becomes a sub one plus four so the second term is going to be the first term 3 plus 4 which is 7. so we have 3 as the first term 7 as a second term so now let's find the next one so let's plug in 3 for n so this becomes a sub 3 the next one this becomes a sub 3 minus 1 or a sub 2 plus 4. so this is seven plus four which is eleven at this point we can see that we have an arithmetic sequence with a common difference of four so to get the next two terms we could just add four it's going to be 15 and 19. so that's it for part a so when dealing with recursive formulas just remember you get your next term by using the previous term now for part b it there's going to be a little bit more work so plugging in n equals 2 we have the second term it's going to be 3 times the first term plus 2. the first term is two so three times two is six plus two that gives us eight so now let's plug in n equals three when n is 3 we have this equation a sub 3 is equal to 3 times a sub 2 plus 2. so we're going to take 8 and plug it in here to get the third term so it's 3 times 8 plus 2 3 times 8 is 24 plus 2 that's 26. now let's focus on the fourth term when n is 4. so this is going to be a sub 4 is equal to 3 times a sub 3 plus 2. so now we're going to plug in 26 for a sub 3. so it's 3 times 26 plus 2. 3 times 26 is 78 plus 2 that's going to be 80. now let's focus on the fifth term so a sub 5 is going to be 3 times a sub 4 plus 2 so that's 3 times 80 plus 2 3 times 8 is 24 so 3 times 80 is 240 plus 2 that's going to be 200 so the first five terms are 2 8 26 80 and 242 so this is neither an arithmetic sequence nor is it a geometric sequence number six write a general formula or explicit formula which is the same for the sequences shown below in order to write a general formula or an explicit formula all we need is the first term and the common difference if it's an arithmetic sequence which for part a it definitely is so if we subtract 14 by 8 we get 6 and if we subtract 20 by 14 we get 6. so we can see that the common difference is positive 6 and the first term is 8. so the general formula is a sub n is equal to a sub 1 plus n minus 1 times d so all we need is the first term and the common difference and we can write a general formula or an explicit formula the first term is eight d is six now what we're going to do is we're going to distribute six to n minus one so we have six times n which is six n and then this will be negative six next we need to combine like terms so eight plus negative six or eight minus six that's going to be positive two so the general formula is six n plus two so if we were to plug in one this will give us the first term eight six times one plus two is eight if we were to plug in four it should give us the fourth term twenty six six times four is twenty four plus two that's twenty six so now that we have the explicit formula for part a what about the sequence in part b what should we do if we have fractions if you have a fraction like this or a sequence of fractions and you need to write an explicit formula try to separate it into two different sequences notice that we have an arithmetic sequence if we focus in the numerator that sequence is two three four five and six for the denominator we have the sequence three five seven nine eleven so for the sequence on top the first term is two and we can see that the common difference is one the numbers are increasing by one so using the formula a sub n is equal to a sub one plus n minus one times d we have that a sub one is 2 and d is 1. if you distribute 1 to n minus 1 you're just going to get n minus 1. so we can combine 2 and negative 1 which is positive 1. so we get the formula n plus one and you could check it when you plug in one one plus one is two so the first term is two if you were to plug in five five plus one is six that will give you the fifth term which is six now let's focus on the sequence of the denominators the first term is three the common difference we could see is two five minus three is two seven minus five is two so using this formula again we have a sub n is equal to a sub one a sub one is three plus n minus one times d d is two so let's distribute two to n minus one so that's gonna be two n minus two and then let's combine like terms three minus two is positive one so a sub n is going to be two n plus one so if we want to calculate the first term we plug in one for n two times one is two plus one it gives us three if we wanna calculate the fourth term and it's four two times four is eight plus one it gives us nine so you always want to double check your work to make sure that you have the right formula so now let's put it all together so we're going to write a sub n and we're going to write it as a fraction the sequence for the numerator is n plus one the sequence for the denominator is two n plus one so this right here represents the sequence that corresponds to what we see in part b and we can test it out let's calculate the value of the third term so let's replace n with three it's going to be three plus one over two times three plus one three plus one is four two times three is six plus one that's seven so we get four over seven if we wish to calculate the fifth term it's going to be five plus one over two times five plus one five plus one is six two times five is ten plus one that's 11. and so anytime you have to write an explicit formula given a sequence of fractions separate the numerator and the denominator into two different sequences hopefully they're both arithmetic if it's geometric you may have to look at another video that i'm going to make soon on geometric sequences but break it up into two separate sequences and then write the formulas that way and then put the two formulas in a fraction and that's how you can get the answer number seven write a formula for the nth term of the arithmetic sequences shown below so writing a formula for the f term is basically the same as writing a general formula for the sequence or an explicit formula so we need to identify the first term which we could see as 5 and the common difference 14 minus 5 is 9 23 minus 14 is 9 as well so once we have these two we can write the general formula so let's replace the first term a sub 1 with 5. and let's replace d with nine now let's distribute nine to n minus one so we're gonna have nine n minus nine next let's combine like terms so it's going to be 9n and then 5 minus 9 is negative 4. so this is the formula for the nth term of the sequence now let's do the same for part b so the first term is 150 the common difference is going to be 143 minus 150 which is negative seven to confirm that if you subtract 136 by 143 you also get negative seven now let's plug it into this formula to write the general equation so a sub n is going to be 150 plus n minus 1 times d which is negative seven so let's distribute negative seven to n minus one so it's going to be 150 minus seven n and then negative seven times negative one that's going to be positive 7. so a sub n is going to be negative 7n plus 157 or you could just write it as 157 minus 7n so that is the formula for the nth term of the arithmetic sequence now let's move on to part b calculate the value of the tenth term of the sequence so we're looking for a sub 10. so let's plug in 10 into this equation so it's gonna be nine times ten minus four nine times ten is ninety ninety minus four is eighty-six so that is the tenth term of the sequence in part a for part b the tenth term is going to be 157 minus seven times ten seven times ten is seventy one fifty seven minus seventy is going to be eighty seven now let's move on to part c find the sum of the first 10 terms so in order to find the sum we need to use this formula s sub n is equal to the first term plus the last term divided by 2 times the number of terms so if we want to find the sum of the first 10 terms we need a sub 1 which we know it's 5. a sub n n is 10 so that's a sub 10 the tenth term is 86 divided by 2 times the number of terms which is 10. 5 plus 86 is 91. 91 divided by 2 gives us an average of 45.5 of the first and last number and then times 10 we get a total sum of 455. so that is the sum of the first 10 terms of this sequence now for part b we're going to do the same thing calculate s sub 10 the first term a sub 1 is 150 the tenth term is eighty seven divided by two times the number of terms which is ten one fifty plus eighty seven that's two thirty seven divided by 2 that's 118.5 times 10 we get a sum of 11.85 so now you know how to calculate the value of the m term and you also know how to find the sum of a series number eight find the sum of the first 300 natural numbers so how can we do this the best thing we can do right now is write a series zero is not a natural number but one is so if we write a list one plus two plus three and this is going to keep on going all the way to 300 so to find the sum of a partial series we need to use this equation s sub n is equal to a sub 1 plus a sub n over 2 times n now let's write down what we know we know that a sub 1 the first term is one we know n is 300 if this is the first term this is the second term this is the third term this must be the 300th term so we know n is 300 and a sub n or a sub 300 is 300 so we have everything that we need to calculate the sum of the first 300 terms so it's a sub 1 which is 1 plus a sub n which is 300 over 2 times the number of terms which is 300 so it's going to be 301 divided by 2 times 300 and that's 45 150. so that's how we can calculate the sum of the first 300 natural numbers in this series number nine calculate the sum of all even numbers from two to one hundred inclusive so let's write a series two is even three is odd so the next even number is four and then six and then eight all the way to one hundred so we have the first term the second term is four the third term is sixty one hundred is likely to be the 50th term but let's confirm it so what we need to do is calculate n and make sure it's 50 and not 49 or 51. so we're going to use this equation to calculate the value of n so a sub n is a hundred let's replace that with a hundred a sub one is two the common difference we see four minus two is two six minus four is two so the common difference is two in this example and our goal is to solve for n so let's begin by subtracting both sides by two a hundred minus two is ninety-eight and this is going to equal two times n minus one next we're going to divide both sides by two 98 divided by two is 49 so we have 49 is equal to n minus one and then we're going to add one to both sides so n is 49 plus 1 which is 50. so that means that 100 is indeed the 50th term so we know that n is 50. so now we have everything that we need in order to calculate the sum of the first 50 terms so let's begin by writing out the formula first so the sum of the first 50 of terms is going to be the first term which is 2 plus a sub 50 the last term which is 100 divided by 2 times n which is 50. so 2 plus 100 that's 102 divided by 2 that's 51. 51 times 50 is 2550. so that is the sum of all of the even numbers from 2 to 100 inclusive try this one determine the sum of all odd integers from 20 to 76 20 is even but the next number 21 is odd and then 23 25 27 all of that are odd numbers up until 75 so a sub 1 is 21 in this problem the last number a sub n is 75 and we know the common difference is two because the numbers are increased by two what we need to calculate is the value of n once we could find n then we could find the sum from 21 to 75. so what is the value of n so we need to use the general formula for an arithmetic sequence so a sub n is 75 a sub 1 is 21 and the common difference is 2. so let's subtract both sides by 21 75 minus 21 this is going to be 54. dividing both sides by 2. 54 divided by 2 is 27. so we get 27 is n minus 1 and then we're going to add 1 to both sides so n is 28 so a sub 28 is 75 75 is the 28th term in the sequence so now we need to find the sum of the first 28 terms it's going to be a sub 1 the first term plus the last term or the 28th term which is 75 divided by 2 times the number of terms which is 28 21 plus 75 that's 96 divided by 2 that's 48 so 48 is the average of the first and the last term so 48 times 28 that's 1 344. so that is the sum of the first 28 terms you