Prove bv induction the standard formula for the sum of the f

Prove bv induction the standard formula for the sum of the first n + 1 terms of a geometric series with ratio r 1: ar k = a(r n+1 - 1)/r - 1.

Statement : a + ar + ar^2 + ....+ ar^n = a(r^(n+1) -1) /(r-1)

a = a(r-1)/(r-1)

the statement is true for n=1...............(1)

let the statement be true for n=k

a + ar + ar^2 + ..+ ar^k = a(r^(k+1)-1)/ (r-1)

adding both side ar^(k+1)

a + ar + ..+ ar^k + ar^(k+1) = a(r^(k+1)-1)/ (r-1) + ar^(k+1)

= a[r^(k+1) -1 + (r-1)r^(k+1)]/(r-1)

= a[r^(k+1) -1 + r^(k+2) -r^(k+1)]/(r-1)

= a[r^(k+2) -1]/(r-1)

=>
a + ar + ..+ ar^k + ar^(k+1) = a[r^(k+2) -1]/(r-1)

statement is true for n= k+1..................(2)

from statements (1), (2)

the statement is proved from induction