Induction Question (1 Viewer)

[Dreamerish*~]

I haven't done induction since year 11.

Prove by mathematical induction that 5²ⁿ - 1 is divisible by 6 when n is a positive integer.

 

[acmilan]

Test for n = 1, 5² - 1 = 24 which is divisible by 6.

Assume true for some n = k.

ie. 5^2k - 1 = 6M (where M is some unknown positive integer)

Need to prove true for n = k+1

5^2(k+1) - 1
= 5^2k+2 - 1
= 5^2k.5² - 1
= (6M + 1).25 - 1 (since 5^2k - 1 = 6M, hence 5^2k = 6M + 1)
= 25.6M + 25 - 1
= 25.6M + 24
= 6(25M + 4)
= 6N (where N = 25M + 4 is an integer since M is an integer and hence 25M + 4 must be an integer)

Hence n = k + 1 is also divisible by 6.

you can then conclude its true for all positive integers n