Question

Use the mathematical induction to prove that 7^n -1 is divisible by 6 whenever n is a positive integer

Answer 1

Explanation:

1) first of all, let s check for n = 1

`7^1 -1=7-1=6`

that s true

2) We assume that this is true for n

`7^n-1` is divisible by 6

what about
`7^(n+1)-1` ?

we know that there is a k natural so that
`7^n-1=6k`

so
`7^n = 1+6k`

then
`7^(n+1) = 7*7^n = 7(1+6k)\\`

so
`7^(n+1)-1 = 7(1+6k)-1 = 6+7*6k = 6(1+7k)`

so it means that
`7^(n+1)-1` is divisible by 6

3) finally as this is true for n=1 and if this is true for n then it is true for n+1 we can conclude that
`7^n-1` is divisible by 6 for n positive integer