Question

Showing all working, find a formula for the general term of the sequence (sn) = s3, s4, s5 . . . defined by

s3 = π and sn = sn-1 - 2 for n >= 4.

Answer 1

An arithmetic progression is a sequence of the form

$$a, a + d, a + 2d, … , a + nd, …$$

where the initial term $a$ and the common difference $d$ are real numbers.

We have

$$a=s_3=\pi,$$

$$d=s_n-s_{n-1}=s_{n-1}-2-s_{n-1}=-2, n\geq4$$

Hence we have the arithmetic progression $\{s_n\}$

$$\pi, \pi+(-2), \pi+2(-2), \pi+3(-2),...,\pi+(n-3)(-2),...$$

The general term of the sequence is

$$s_n=\pi-2(n-3), n\ge 3$$