Can someone help me prove the below by induction please? Thanks :)

Question

Prove by induction:

n
Σ(q^i) = (q^(n+1)-1)/(q-1)
i=0

Answer 1

Proof by induction:
(1) prove the initial case
(2) show that if it is true through for n, it is also true for n+1
So (1) demonstrate for n =0
Evaluate proposed right hand side for n=0

(q^1-1)/(q-1)
(q-1)/(q-1) =1
Left hand side
Q^0 =1
So it is true that the left hand side equals right hand side for n=0
(2) evaluate both sides for n+1 by adding the term q^(n+1)
Left hand side:
(Sum up to n+1) = (sum up to n) + q^(n+1)
Right hand side
q^(n+1)-1)/(q-1) + q^(n+1)
Combine over common denominator
((q^(n+1)-1) + q^(n+1)*(q-1)) /(q-1)
((q^(n+1)*(q-1+1) -1)) /(q-1)
((q^(n+1)*q -1))/(q-1)
((q^(n+2) -1))/(q-1)
This is the right hand side for the n+1 case
So the left hand side for n+1 equals the right hand side for n+1. So the induction step is true.