Question

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

Answer #1

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

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

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

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

Exercise 6.6. Let the inductive set be equal to all natural
numbers, N. Prove the following propositions. (a) ∀n, 2n ≥ 1 +
n.
(b) ∀n, 4n − 1 is divisible by 3.
(c) ∀n, 3n ≥ 1 + 2 n.
(d) ∀n, 21 + 2 2 + ⋯ + 2 n = 2 n+1 − 2.

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

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

Prove by induction on n that 13 | 2^4n+2 + 3^n+2 for all natural
numbers n.

Find all natural numbers n so that
n3 + (n + 1)3 > (n +
2)3.
Prove your result using induction.

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