Question

Given the statement 11^(n -6) is divisible by 5 for every positive integer n. What must be shown in the induction step?

Answer 1

Answer:

Proved

Step-by-step explanation:

The given data is:

$11^n - 6$ not $11^{n - 6$

Required

Prove by induction that it is divisible by 5

Assume $n = k$

So, we have:

$11^k - 6 = 5p$ where p is a positive digit

Rewrite as:

$11^k = 5p + 6$

To solve further, we have to prove that $11^k - 6$ is true for $n=k+1$

So, we have:

$11^{k+1} - 6$

$11^{k+1} - 6 = 11^k * 11 - 6$

Add 0

$11^{k+1} - 6 = 11^k * 11 - 6 + 0$

Replace 0 with $-5 +5$

$11^{k+1} - 6 = 11^k * 11 - 6 - 5 + 5$

$11^{k+1} - 6 = 11^k * 11 - 11- 5$

Factorize

$11^{k+1} - 6 = 11 (11^k - 1)- 5$

Substitute $11^k = 5p + 6$

$11^{k+1} - 6 = 11 (5p + 6- 1)- 5$

$11^{k+1} - 6 = 11 (5p + 5)- 5$

Factorize

$11^{k+1} - 6 = 11 *5(p + 1)- 5$

Factorize

$11^{k+1} - 6 = 5[11(p + 1)- 1]$

The 5 outside the bracket implies that it is divisible by 5