How do you solve this question?

Question

How many positive values of n are there such that the geometric series 1 + 2 + 2^2 + ... +2^n is divisible by 9?

Please help me on this. Thanks!

Answer 1

let us evaluate this upto n terms

we get 2^(n) -1


now this to be divisible by 9

2^n mod 9 = 1

2^1 mod 9 = 2
2^2 mod 9 = 4
2^ 3 mod 9 = 8 or -1

so 2^6 mod 9 = 1

so (2^6n)
so 2^6-1 mod 9 = 0

so 6k terms sum is dvisible by 0

so n = 6, 12 , 18 so on of the form 6k

Answer 2

The sum of that series is equal to 2^(n+1) - 1

So we want to know what values of n will make that sum divisible by 9.

It seems from experimenting that when n is any multiple of 5, the sum will be divisible by 9.