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 2. To go from the second term to the third term, you need to multiply by 2 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 3 and 11, 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 were to 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 3 and 6. 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 would 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 6 times 16. 6 times 6 is 36. The square root of 36 is 6. The square root of 16 is 4. So we have 6 times 4, which is 24. So as you can see, the geometric mean of two numbers within a 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, it's a sub n is equal to a1 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 3, n is 5 since we're looking for the fifth term.
The common difference is 4 in this problem. 5 minus 1 is 4, 4 times 4 is 16, 3 plus 16 is 19. So this formula gives you any term in the sequence. You can find the 5th term, the 7th term, the 100th term, and so forth.
Now, in a geometric sequence, we could use this formula. So let's calculate the sixth term of the geometric sequence. It's going to be a sub 6, which equals a sub 1. The first term is 3. The common ratio is 2. And this is going to be raised to the 6 minus 1. 6 minus 1 is 5. And then 2 to the fifth power. If you multiply 2 five times, 2 times 2 times 2 times 2 times 2. So we can write it out.
So this here, that's 4. 3 2's make 8. 4 times 8 is 32. So this is 3 times 32. 3 times 30 is 90. 3 times 2 is 6. So this will give you 96. So that's how you can find the nth 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 2 times n. For a 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 the last term is 15 times 7. 10 times 7 is 70. 5 times 7 is 35. So this is going to be 105. So that's the sum of the first seven terms. And you can confirm this with your calculator.
If you add up 3 plus 7 plus 11 plus 15 plus 19 plus 23 and then plus 27, 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 6. So this is going to be 3 plus 6 plus 12 plus 24. 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 1, which is 3. times 1 minus r, r is the common ratio, which is 2, raised to the n, n is 6, over 1 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 5th 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 can 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've 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 there's also a finite series and an infinite series.
This sequence is finite. It has a beginning and it has an end. 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 3 plus 7 plus 11 plus 15 plus 19 and then plus dot 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 a 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. in 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 3. 4 plus 3 is 7. 7 plus 3 is 10. 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 2. 4 times 2 is 8. 8 times 2 is 16. So we have a common ratio, which makes this sequence geometric. For answer choice C, going from 5 to 9, that's plus 4. And from 9 to 13, that's plus 4. 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 2 to 6, we're multiplying by 3. And then 6 times 3 is 18. So that's a geometric. the geometric series. Now for E, going from 50 to 46, that's a difference of negative 4, and 46 to 42, that's a difference of negative 4. So this is arithmetic. For F, we have a common ratio of 4. 3 times 4 is 12. 12 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 is 3. Or you can 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. It's 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 6. 24 minus 18 gives us the same common difference of 6. So this is going to be a 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, 2 to the 3rd is 8, which means that we have 8 different possible combinations. Right now, I have 7 out of the 8 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 a series if it's arithmetic or 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 3n minus 7. So the first thing we're going to do is find the first term.
So we're going to replace n with 1. So it's going to be 3 minus 7, 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. 3 times 3 is 9, minus 7, that's 2. And then the fourth term, a sub 4, that's going to be 12 minus 7, which is 5. So we have a first term of negative 4, then it's negative 1, 2, 5. And then the sequence can continue. So the common difference in this problem is positive 3. Going from negative 1 to 2, if you add 3, you'll get 2. And then 2 plus 3 is 5. But this is the answer for the problem. So those are the first four terms of the sequence. Number 3. 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 7. So we have a common difference of 7. So we could use that to find 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 a1 and d.
So we know the first term is 29 and the common difference is negative 4. So this is all we need to write the first five terms. If the common difference is negative 4, then the next term is going to be 29 plus negative 4, which is 25. 25 plus negative 4, 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, what 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 2, this is a sub 2, and that's going to equal a sub n minus 1. 2 minus 1 is 1, so this becomes a sub 1 plus 4. 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 the 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 7 plus 4, which is 11. At this point, we can see that we have an arithmetic sequence with a common difference of 4. So to get the next two terms, we can just add 4. 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, 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 2, so 3 times 2 is 6 plus 2, that gives us 8. So now, let's plug in n equals 3. 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 242. 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 in a 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 that's it. 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 8. d is 6. Now what we're going to do is we're going to distribute 6 to n minus 1. So we have 6 times n, which is 6n, and then this will be negative 6. Next, we need to combine like terms. So 8 plus negative 6, or 8 minus 6, that's going to be positive 2. So the general formula is 6n plus 2. So, if we were to plug in 1, this will give us the first term, 8. 6 times 1 plus 2 is 8. If we were to plug in 4, it should give us the fourth term, 26. 6 times 4 is 24, plus 2, that's 26. 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 on the numerator.
That sequence is 2, 3, 4, 5, and 6. For the denominator, we have the sequence 3, 5, 7, 9, 11. So for the sequence on top, the first term is 2, and we can see that the common difference is 1. The numbers are increasing by 1. So using the formula a sub n is equal to a sub 1 plus n minus 1 times d. We have that a sub 1 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 1. And you can check it. When you plug in 1, 1 plus 1 is 2, so the first term is 2. If you were to plug in 5, 5 plus 1 is 6, that will give you the fifth term, which is 6. Now let's focus on the sequence of the denominators.
The first term is 3. The common difference we could see is 2. 5 minus 3 is 2. 7 minus 5 is 2. So using this formula again, we have a sub n is equal to a sub 1. a sub 1 is 3 plus n minus 1 times d. d is 2. So let's distribute 2 to n minus 1. So that's going to be 2n minus 2. And then let's combine like terms. 3 minus 2 is positive 1. So a sub n is going to be 2n plus 1. So if we want to calculate the first term, we plug in 1 for n, 2 times 1 is 2, plus 1, it gives us 3. If we want to calculate the fourth term, n is 4, 2 times 4 is 8, plus 1, it gives us 9. 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 1. The sequence for the denominator is 2n plus 1. 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 3. It's going to be 3 plus 1 over 2 times 3 plus 1. 3 plus 1 is 4. 2 times 3 is 6 plus 1. That's 7. So we get 4 over 7. If we wish to calculate the fifth term, it's going to be 5 plus 1 over 2 times 5 plus 1. 5 plus 1 is 6. 2 times 5 is 10 plus 1. 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 nth 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 can 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 9. Now let's distribute 9 to n minus 1. So we're going to have 9n minus 9. 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 f 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 7. To confirm that, if you subtract 136 by 143, you also get negative 7. 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 7. So let's distribute negative 7 to n minus 1. So it's going to be 150 minus 7n, and then negative 7 times negative 1, 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...
57 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 10th term of the sequence. So we're looking for a sub 10. So let's plug in 10 into this equation. So it's going to be 9 times 10 minus 4. 9 times 10 is 90 90 minus 4 is 86 so that is the 10th term of the sequence in Part A for Part B the 10th term is going to be 157 minus 7 times 10 7 times 10 is 70 157 minus 70 it's going to be 87 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 10th 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 87, divided by 2, times the number of terms, which is 10. 150 plus 87, that's 237, divided by 2, that's 118.5, times 10, we get a sum of 1185. So now you know how to calculate the value of the nth term, and you also know how to find the sum of a series. Number 8. Find the sum of the first 300 natural numbers. can we do this?
The best thing we can do right now is write a series. 0 is not a natural number, but 1 is. So if we write a list, 1 plus 2 plus 3, 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 1. 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 9. Calculate the sum of all even numbers from 2 to 100 inclusive. So let's write a series. 2 is even, 3 is odd, so the next even number is 4, and then 6, and then 8, all the way to 100. So we have the first term. The second term is 4. The third term is 60. 100 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 and 51. So we're going to use this equation to calculate the value of n. a sub n is 100. Let's replace that with 100. a sub 1 is 2. The common difference We can see 4 minus 2 is 2. 6 minus 4 is 2. So the common difference is 2 in this example. And our goal is to solve for n. So let's begin by subtracting both sides by 2. 100 minus 2 is 98. And this is going to equal 2 times n minus 1. Next, we're going to divide both sides by 2. 98 divided by 2 is 49. So we have 49 is equal to n minus 1. And then we're going to add 1 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 1. 102 divided by 2, that's 51. 51 times 50 is 2,550.
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 2, because the numbers are increasing by 2. What we need to calculate is the value of n. Once we can find n, then we can 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.