Prove by induction: "For all \( n \in \mathbb{N} \), \( \sum_{k=1}^{n} k \cdot k! = (n+1)! - 1 \).

Prove by induction that for positive integers n:  4|(5^(n+1)+ 3^(2n-1)).

1 and 2an + 2" for all integers n, n > 1. Use induction to prove that 2. (4 points) Suppose {an} is a sequence recursively defined by a1 = ап+1 || an = n2"-' for all positive integers n.

Use mathematical induction to prove the inequality n! < n^n for all positive integers n ≥ 2.

Prove the statement by mathematical induction. (1- (1/2^2)) (1-(1/3^2))... (1-(1/n^2)) = (n+1)/(2n), for every integer n ≥ 2.

Use mathematical induction to prove that for every natural number, say n: 1/2 + 1/6 + 1/12 + ... + 1/(n^2+n)=n/(n+1)

Prove by induction (write an original proof) For all integers N ≥ 1, 2 + 6 + 10 +...+ (4N-2) = 2N^2

For all n greater than or equal to 1, prove the following by mathematical induction: 1/(1^2) +1/(2^2) + 1/(3^2) .... + 1/(n^2) smaller than or equal to 2-(1/n).

Prove the following using mathematical induction: For all natural numbers n, 3^(n+2) > (n + 3)^2.

Prove that by mathematically Induction. 2^n < (n + 1)!, for all integers n ≥ 2.