Question

If `a_1` and r are integers, why is the value of `(a_1 - a_1r^n)/(1-r)` must also be an integer?

Answer 1

See below

Explanation:

`(1-r^n)/(1-r) = 1+r+r^2+...+r^(n-1)`
This is certainly an integer for integer values of `r` and `n`. So

`a_1(1-r^n)/(1-r)`

has to be an integer for integer `a_1` and `r`

Alternatively :

`1-r^n` vanishes for `r=1`. So iy must be divisible by `1-r`. Hence `(1-r^n)/(1-r)` is an inetger.