Question 429173: Consider the sequence 8, 6, 3, –1, –6, ...
a. Find the next two terms of the sequence.
b. Write an explicit formula for the sequence.
c. Write a recursive formula for the sequence.
I got part a, but I'm not sure how to do the other two parts, considering that the sequence in neither arithmetic or geometric.
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! a. The n-th term in the series is obtained by subtracting n from the previous term. So the 6th term is -6 - 6 = -12, and the 7th term is -12 - 7 = -19.
c. The recursive formula which gives the n-th term from the (n-1)-th term is:
a_n = a_n-1 - n
b. To get an explicit formula, we need to find an expression which gives the n-th term without having to compute earlier terms in the sequence. Looking at the numbers, and from the recursive formula, we see that the sequence is built by subtracting n from the previous term. This is similar to the triangular number sequence 1,3,6,10,15,... which has the explicit formula a_n = n(n+1)/2. In our case we are subtracting n from the previous term, so we multiply by -1/2 instead of 1/2. However, we also need to add a constant term to reproduce the numbers of the sequence. We can write a_1 = -1(2)/2 + c = 8. Therefore, c = 9.
So the explict formula is:
a_n = -n(n+1)/2 + 9
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