Prove that for every n 1 Prove that for every n 1 31 32 3

Prove that for every n ? 1,

Prove that for every n 1, 3^1 + 3^2 + 3^3 + ... + 3^n = 3^n + 1 - 3 / 2

Solution

lets prove by induction

S(n) ; 3 + 3^2 +...3^n = (3^(n+1) -3)/2

3 = (3^2-3)/2

=>
S(1) is true

let S(k) is true

=>

3 + 3^2 + ....+3^k = [3^(k+1) -3]/2

=>

3 + 3^2 + ....+3^k + 3^(k+1) = [3^(k+1) -3]/2 + 3^(k+1)

=>

3 + 3^2 + ....+3^k + 3^(k+1) = [3^(k+1) -3]/2 + 3^(k+1) = [3*3^(k+1) -3]/2

=>

3 + 3^2 + ....+3^k + 3^(k+1) = [3^(k+2) -3]/2

=>

S(k+1) is true

thus proved by induction