Which formula does NOT describe the sequence 8, 6, 3, - 1, - 6, ...?

Where is the rest of the question

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