Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.4 + 9 + 14 + ... + (5n - 1) = (5n + 3)

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 4 = (5(1) + 3) = 4.
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.
4 + 9 + 14 + ... + (5k - 1) + 5(k + 1) - 1 = (5k + 3) + 5(k + 1) - 1
 = [k(5k + 3) + 10(k + 1) - 2]
 = (5k² + 3k + 10k + 10 - 2)
 = (5k² + 13k + 8)
 = (k + 1)(5k + 8)
 = (5k + 5 + 3)
 = (5(k + 1) + 3)
Condition II is satisfied. As a result, the statement is true for all natural numbers n.