In Exercises 23–34, write a formula for the general term (the nth...

Question

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. an= an-1 -10, a₁ = 30

Accepted Answer

Hello. Today we're going to be using the given equation to find the equation that describes the end term of the arithmetic sequence. Then we're going to use that equation to find the 40th term of the sequence. Now, in order to find the general equation of the end term, we need to understand the general form which is a sub N. Is equal to the first term, A sub one plus the quantity and minus one times D. Now d represents our common difference and a sub one represents the first term of the given sequence in the problem we are given the first term of the sequence which is equal to 20. So a sub one is equal to 20. Now we have a sub one but we're missing our common difference D. While we have the general form of the recurrence formula that's going to help us solve for D. And that is labeled as a sub N. Is equal to a sub n minus one plus D. Now in the problem here we are given a recurrence formula, we are told that a sub N is equal to a sub n minus one plus 12. So if we compare that to the general form of the recurrence formula, as you can see, 12 takes up the form of D. So here we can allow D two equal to 12. Now that we have the two substitution, we need to plug into the general equation of the N term. We can now find the equation that describes the end term of the arithmetic sequence. So here a sub N is equal to a sub one which is equal to 20 plus the quantity and minus one times D. Which we said is 12. Now we just need to go ahead and simplify this in order to simplify we're going to distribute 12 into the quantity and minus one and doing so is going to give us a sub N is equal to 20 plus 12 N minus 12. And now we just need to go ahead and combine like terms 20 minus 12 is going to give us eight. So we have a sub N is equal to eight plus 12 N. And this is going to be the answer to the first part of this question because we are asked to find the general equation that describes the end term of the arithmetic sequence. Now that we have the equation we are asked to find the 40th term of the sequence. So since we are asked to find the 40th term of the sequence, we're going to allow N to equal the 40 and we're going to take this value of N and plug it into the equation that we just found. So doing so is going to give us a sub 40 is equal to eight plus 12 times 40. Now 12 times 40 is going to give us And 480-plus 8 is going to give us 488. So that means that the 40th term a sub 40 is equal to 488 and that's gonna be the other part of the answer that we are looking for. So just to recall, the general equation of the end term is a sub N is equal to eight plus 12 N. And the 40th term of the sequence is going to be 488. With that being said, the answer to this problem is going to be be So I hope this video helps you in understanding how to find the equation of the 10th term. And I'll go ahead and see you all in the next video.

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. an= an-1 -10, a₁ = 30

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