Prove by induction that f(n)=2^(2n-1)+3^(2n-1) is divisible by 5 for n in ZZ^+?

1 Answer
Jan 28, 2018

See below.

Explanation:

Note that for m odd we have

(a^m+b^m)/(a+b) = a^(m-1)-a^(m-2)b+ a^(m-3)b^2 + cdots -a b^(m-2)+b^(m-1)

which demonstrates the afirmation.

Now by finite induction.

For n = 1

2+3 = 5 which is divisible.

now supposing that

2^(2n-1)+3^(2n-1) is divisible we have

2^(2(n+1)-1)+3^(2(n+1)-1) =2^(2n-1) 2^2+3^(2n-1)3^2=
= 2^(2n-1) 2^2+3^(2n-1)2^2+5 xx 3^(2n-1)=

= 2^2(2^(2n-1)+3^(2n-1))+5 xx 3^(2n-1) which is divisible by 5

so it is true.