Show that `7^(2n-1) - 1` is divisible by 48

Expert Answers
justaguide eNotes educator| Certified Educator

It has to be determined if `7^(2n - 1) - 1` is divisible by 48.

For n = 1, `7^(2n - 1) - 1 = 48`

Let `7^(2n - 1) - 1` be divisible for any integer n,

`7^(2n - 1) - 1 = 48*k` , where k is an integer

The value of the expression for n + 1 is:

`7^(2(n+1) - 1) - 1 `

= `7^((2n + 2) - 1) - 1`

= `7^(2n + 1) - 1`

= `7^(2n - 1)*7^2 - 1`

Substitute the expression derived for n

= `(48k +1)*49 - 1`

= `48*49*k + 49 - 1`

= `48*49*k + 48`

This proves that `7^(2n - 1) - 1` is divisible by 48 for all `n >= 1`