Zig-zag sequence

Algebra Level 4

Consider the sequence: $65, 42, 67, 44, 69, 46, 71, 48 \, \ldots$

The $n^\text{th}$ term of this sequence can be expressed as $an+b(-1)^n +c$ , where $a,b,c \in \mathbb{Z}$ .

Evaluate $a+b+c$ .

Note: The 1st term of the sequence can be expressed as $a \times 1 + b(-1)^2 + c$ .

1 solution

Chew-Seong Cheong
Nov 12, 2016

It is given that the $n$ th term $x_n = an+b(-1)^n+c$ , therefore the first three terms are:

$\begin{cases} x_1 = 65 & \implies a - b + c = 65 \\ x_2 = 42 & \implies 2a + b + c = 42 \\ x_3 = 67 & \implies 3a - b + c = 67 \end{cases}$

$\begin{aligned} x_3-x_1 & = 67-65 \\ 2a & = 2 \\ \implies a & = 1 \end{aligned}$

$\begin{aligned} x_1+x_2 & = 65+42 \\ 3a + 2c & = 107 \\ 2c & = 107 - 3 \\ \implies c & = 52 \end{aligned}$

$\begin{aligned} x_1 & = 65 \\ a - b + c & = 65 \\ b & = 1 + 52 - 65 \\ \implies b & = -12 \end{aligned}$

$\implies a + b + c = 1-12+52 = \boxed{41}$