Proving Induction Inequality: 5^n+9 < 6^n for n>=2

[nastygoalie89]

Homework Statement

5^n + 9 < 6^n for all integers n>=2.

Homework Equations

The Attempt at a Solution

Induction proof.

Base Case: 5^(2) + 9 < 6^(2)
34<36
P(k): 5^k + 9 < 6^k
P(k+1): 5^(k+1) + 9 < 6^(k+1)

how do i prove p(k) can equal p(k+1)?

 

[Mark44]

Homework Statement

5^n + 9 < 6^n for all integers n>=2.

Homework Equations

The Attempt at a Solution

Induction proof.

Base Case: 5^(2) + 9 < 6^(2)
34<36
P(k): 5^k + 9 < 6^k
P(k+1): 5^(k+1) + 9 < 6^(k+1)

how do i prove p(k) can equal p(k+1)?

You don't prove that P(k) = P(k+1). You assume that statement P(k) is true and use this fact to prove that statement P(k + 1) is also true.

From your induction hypothesis (statement P(k)) you are assuming that
5^k + 9 < 6^k

Work with 5^(k + 1) + 9 and see what you get.