how to calculate 2^n modulo 1000000007 , n = 10^9

Question

what is the fastest method to calculate this, i saw some people using matrices and when i searched on the internet, they talked about eigen values and eigen vectors (no idea

Answer 1

f(n) = (2*f(n-1)) + 2 with f(1)=1

is equivalent to

(f(n)+2) = 2 * (f(n-1)+2)
         = ...
         = 2^(n-1) * (f(1)+2) = 3 * 2^(n-1)

so that finally

f(n) = 3 * 2^(n-1) - 2

where you can then apply fast modular power methods.