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

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Answer 1 · 2018-04-08T18:07:49

See below

$(1-r^n)/(1-r) = 1+r+r^2+...+r^(n-1)$
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.

If a_1a_1 and r are integers, why is the value of (a_1 - a_1r^n)/(1-r)(a_1 - a_1r^n)/(1-r) must also be an integer?