Mathematical Induction: 2 problems

Question

Use mathematical induction to prove the following statements. Show your work.

 

1.    3 + 6 + 9 + ... + 3n = (3n(n + 1))/2

 

2.    4 + 4^2 + 4^3 + ... + 4^n = 4/3 ((4^n) - 1)

Answer 1

It is a proven theorem that the sum of the first n integers, 1+2+3=...=n = n(n+1)/2

The first statement is merely the same, but 3-fold

The second statement holds for n=1 since 4 = (4/3)(3) = (4/3) ( 4^1 -1)

Given: 4 + 4^2 + 4^3 + ..... + 4^n = (4/3)( 4^n - 1)

4 + 4^2 + 4^3 + ..... + 4^n + 4^(n+1) =

(4/3)(4^n - 1) + 4^(n+1) =

(1/3) * 4^(n+1) - 4/3 + 1*4^(n+1)

=(4/3)* 4^(n+1) - 4/3

= (4/3) [ 4^(n+1) - 1]