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

1. Define 0! = 1. For any positive integer n, define n! = n. (n-1)!. For example, 6! = 6 x 5 x 4 x 3×2×1=720. 4" Use induction to show that (2n)! for all integern> 2. (n!)2 n+1

Please use strong induction to prove that each integer n that is greater than 0 can bewritten as a sum of different non-negative integer powers of 2.Hint: try to use a formula to represent ”sum of different non-negative integer powers of 2” beforestarting to write your proof, for example, 9 = 20 + 23 and 14 = 21 + 22 + 23

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

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.

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).

Determine using induction all the positive integer values of n for n^3 > 2^n. Prove your answer using mathematical induction.    Please read the question. This is NOT asking to prove that all integers satisfy this, it is asking to find the integers that do satisfy it via induction.  Please explain every step.

Prove by induction that n! < n^n for all n >= 2  Please inlcude details. Thank you!

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

Prove by induction the attachment for all n >= 2.