Find the sum of the first 20 terms of the arithmetic sequence:...

Question

Find the sum of the first 20 terms of the arithmetic sequence: 4, 10, 16, 22,……….

Accepted Answer

Hello. Today we are asked to find the sum of the 1st 40 terms of the given arithmetic sequence. Now we have an equation that allows us to find the sum of any arithmetic sequence. And that is when s is equal to end over two times the quantity a sub one plus a sub N. Here and represents the amount of terms that are within your arithmetic sequence. A sub one represents the first term in your arithmetic sequence and a sub N represents the last term of the arithmetic sequence. So the sequence given to us is 39 1521. All the way to the 40th term. Now we are told that we want to find the sum of the 1st terms. So we're going to allow N2 equal to 40 since that's the amount of terms that within our arithmetic sequence a sub one represents the first term of our arithmetic sequence and here our first term is three. So a sub one is going to equal to three. Now we need to find the last term which is represented by a sub N. However we are not given the last term in this arithmetic sequence. So we're going to need to solve for it. We have an equation that allows us to find any term with an arithmetic sequence and that is given as a sub N is equal to the first term. A sub one plus the quantity and minus one times the common difference which is D. Now let's go ahead and figure out what the common difference is to go from 3-9. We need to add a total of six And to go from 9 to 15 we're adding a total of 6-9 and to go from 15 to 21 we're adding 6-15. So every time we increase the number we are adding six to every term. So the common difference D. Is going to equal to six. So we have a sub one which is three and we have D. Which is six. Let's go ahead and plug this into the equation in order to figure out what the equation is for the end term. So that's going to be a sub N is equal to the first term which is three plus the quantity and minus one times the common difference which is six. Now what we want to go ahead and do is simplify this. In order to simplify we're going to distribute six into the quantity and minus one. This is going to give us a sub N is equal to three plus six N minus six. And now we're going to combine like terms three minus six is going to give us negative three. So the general term in this case is going to be a sub N is equal to negative three plus six. N. Now we need to figure out what the 40th term is. We're going to allow N two equal to 40. So what that does is give us a sub 40 is equal to negative three plus six times 40 six times 40 Is going to give us 240 so we have a sub 40 is equal to negative 3-plus and 240 -3 is going to give us 237. So the 40th term in this case is going to be a sub 40 which is equal to 237. Now we can take this and substitute it in for a sub N in our some equation. So now we have all the substitution is we need to find the sum of the arithmetic sequence. So let's go ahead and solve for it S is going to equal to n which is 40/2 times a sub one which is three plus the last term, which is the 40th term which is 237 divided by two is going to give us 20 And plus three is going to give us 240. Finally 240 times 20 is going to give us 4800. So the sum is going to be 4800 which is the answer to this given arithmetic sequence. And that means that the answer to this problem is going to be be So I hope this video helps you in understanding how to find the sum of an arithmetic sequence and I'll go ahead and see you all in the next video.

Find the sum of the first 20 terms of the arithmetic sequence: 4, 10, 16, 22,……….

Key Concepts

Arithmetic Sequence

Common Difference

Sum of an Arithmetic Sequence

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