Question

Prove by mathematical induction that the sum of the first n terms of the geometric series 1/5 + 1/5^2 + ... is 1/4(1-1/5^n+1)

Answer 1

We know from the partial sum formula for a geometric series that
Sn = 1/5 (1 - 1/5^n)/(1 - 1/5) = 1/4 (1 - 1/5^n)
So I guess your formula is incorrect.
But we can still use induction.
test n=1: 1/5 = 1/4 (1 - 1/5)
assume it holds for n=k, and test for n=k+1
1/5 + ... + 1/5^k + 1/5^(k+1) = 1/4 (1 - 1/5^k) + 1/5^(k+1)
= 1/4 (1 - 1/5^k) + 1/5 * 1/5^k
= 1/4 + (1/5 - 1/4) * 1/5^k
= 1/4 - 1/20 * 1/5^k
= 1/4 (1 - 1/5 * 1/5^k)
= 1/4 (1 - 1/5^(k+1))
QED