Sum of a recursive sequence

Algebra Level 4

Let $\left\{ a_{n} \right\}_{n = 1}^{\infty}$ be an integer sequence such that

$a_{n + 2} = \begin{cases} a_{n} + a_{n + 1}, \: \: \text{n is even}\\ \; a_{n} - a_{n + 1}, \: \: \text{otherwise} \end{cases}$

where $a_{1} = a_{2} = 1$ .

Find the sum $S_{100}$ of the first one hundred members.

-1 0 1 2 3 12

1 solution

Saya Suka
Jan 5, 2021

This recursive sequence is cyclic with a period of 12.

It goes : 1,1,0,1,-1,0,-1,-1,0,-1,1,0
with a periodic sum of 0.
100 = 12 x 8 + 4, so
S(100) = 8 x 0 + S(4)
= 0 + (1+1+0+1)
= 3