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 +3, a₁ = 4

Accepted Answer

Hello. Today we are asked to find the equation that describes the end term of the following arithmetic sequence. Then we are asked to use that equation to find the 25th term of the sequence. So since we are asked to find the general form of the term of the sequence, we're going to use the general equation of the term which is a sub N. Is equal to your first term. A sub one plus the quantity and minus one times D. And this is where D represents the common difference in the arithmetic sequence. Now in the problem, we are told what the first term is. We are told that the first term is eight. So we're going to allow a sub one to equal to eight. However, we are missing the value D. Now we do have an equation for the recurrence formula that's going to help us solve for D. The equation of the recurrence formula is a sub N is equal to a sub n minus one plus D. And in the problem here we are given a recurrence formula, a sub N is equal to a sub N -1 -7. So again a sub N is equal to a sub n minus one minus seven. Now notice that negative seven takes the place of D. So we're going to allow D to equal to negative seven. Now we have a substitution for D and we have a substitution for a sub one. We can plug these into the general form of the equation and that will allow us to find the equation that describes the term of the sequence. So we have a seven is equal to a sub one which is eight plus the quantity and minus one times D. Which we said is negative seven. Now we just need to go ahead and simplify this in order to simplify we're going to distribute negative seven to the quantity and minus one and doing so is going to give us a sub N is equal to eight minus seven N plus seven because Native seven times negative one produces a positive seven. And now we just need to go ahead and combine like terms eight plus seven is going to give us 15. So we have a sub N is equal to 15 minus seven N. And this is going to be the equation that describes the end term of our arithmetic sequence. And that's going to be the first part of our answer. Now, since we have this equation, we are asked to find the 25th term of the arithmetic sequence. Since we are looking for the 25th term, we're going to allow N to equal the 25 we're going to take this value of N and plug it in to the empty equation And doing so is going to give us a sub 25 is equal to 15 minus seven times 25 Seven times 25 is going to give us the value of 175 and 15 -175 is going to give us -160. So a sub 25 is going to equal to negative 160 and that is going to be the 25th term of the arithmetic sequence. So again, the equation of the end term is a sub N is equal to 15 minus seven N. And the 25th term is going to be negative 160. With that being said, the answer to this problem is going to be C. So I hope this video helps you in understanding how to find the equation of the term of an arithmetic sequence and I'll go ahead and see you all in the next video.

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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 +3, a₁ = 4