Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.3 + 8 + 13 + ... + (5n - 2) = (5n + 1)

Question

What will be an ideal response?

Answer 1

First we show that the statement is true when n = 1.
 For n = 1, we get 3 = (5(1) + 1) = 3.
This is a true statement and Condition I is satisfied.

Next, we assume the statement holds for some k. That is,
  is true for some positive integer k.
We need to show that the statement holds for k + 1. That is, we need to show that
 
So we assume that  is true and add the next term, , to both sides of the equation.
3 + 8 + 13 + ... + (5k - 2) + 5(k + 1) - 2 = (5k + 1) + 5(k + 1) - 2
 = [k(5k + 1) + 10(k + 1) - 4]
 = (5k² + k + 10k + 10 - 4)
 = (5k² + 11k + 6)
 = (k + 1)(5k + 6)
 = (5k + 5 + 1)
 = (5(k + 1) + 1)
Condition II is satisfied. As a result, the statement is true for all natural numbers n.