Question

In an arithmetic sequence, a17 = -40 and
a28 = -73. Explain how to use this information to write a recursive formula for this sequence.

Answer 1

In an arithmetic sequence the common difference d is constant. Compute it from the two given terms:

d = (a28 − a17)/(28 − 17) = (−73 − (−40))/11 = −33/11 = −3.

A recursive formula needs an initial term and the step. You can use a17 as the initial term:
- a17 = −40
- a_n = a_{n−1} − 3 for n > 17.

If you prefer to start at a1, find a1: a17 = a1 + 16d ⇒ −40 = a1 + 16(−3) ⇒ a1 = 8. Then
- a1 = 8
- a_n = a_{n−1} − 3 for n ≥ 2.

(For reference, the explicit formula is a_n = 8 + (n−1)(−3) = 11 − 3n.)