Question

. **Prove that 2^(2n-1) + 3^(2n-1) is divisible by 5 for
every natural number n.**

Answer #1

prove that 2^2n-1 is divisible by 3 for all natural numbers n ..
please show in detail trying to learn.

Prove the following using induction:
(a) For all natural numbers n>2, 2n>2n+1
(b) For all positive integersn,
1^3+3^3+5^3+···+(2^n−1)^3=n^2(2n^2−1)
(c) For all positive natural numbers n,5/4·8^n+3^(3n−1) is
divisible by 19

Prove that for n>=1, (2n-1)^2-1 is divisible by 8.

Prove by mathematical indution: If n is a natural number, then
1/2*3 + 1/3*4 + 1/4*5 +....+1/(n +1)(N+2)=n/2n+4

Prove that for each positive integer n, (n+1)(n+2)...(2n) is
divisible by 2^n

Prove by induction that if n is an odd natural number,
then 7n+1 is divisible by 8.

(10) Use mathematical induction to prove that
7n – 2n is divisible by 5
for all n >= 0.

If n is a natural number, then 1 * 5 + 2 * 6 + 3 * 7 + ------ +
n(n +4) = n(n+1)(2n+13)/6.

Prove that (n − 1)^3 + n^ 3 + (n + 1)^3 is divisible by 9 for
all natural numbers n.

Show by induction that 1+3+5+...+(2n-1) = n^2 for all n in the
set of Natural Numbers

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