136k views
4 votes
Use the mathematical induction to prove that 7^n -1 is divisible by 6 whenever n is a positive integer

User Theadam
by
7.4k points

1 Answer

4 votes

Answer:

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

User Patrick Beeson
by
6.7k points