How many terms are in the arithmetic sequence 5, 1, −3, . . , −111?

hint: an = a1 + d(n − 1), where a1 is the first term and d is the common difference.

Here, it helps to find a general expression for the nth term in the sequence. Since the sequence is arithmetic, the general term is a_n = a + bn, where b is the common difference. By inspection of the first few terms we see the common difference is -4. Then by solving a_1 = a - 4x1 = 5, we obtain a = 9, so the nth term is: a_n = 9-4n. We then solve a_n=-111 for n, so 9-4n=-111. Rearranging, we get 4n=9+111, or 4n=120, or n=30. Thus there are 30 terms in the sequence.