SOLUTION: prove that 【(n+1)^n】+1 is divisible by n^2 by using binomial theorem

Question 1207212: prove that 【(n+1)^n】+1 is divisible by n^2 by using binomial theorem

Answer by Shin123(626) (Show Source): You can put this solution on YOUR website!
Unfortunately, that is not true, since taking n=2 gives that 10 is divisible by 4, which is clearly false.
Did you mean instead? In that case, the statement is true, and the proof is as follows.
To show this, we can expand out with the binomial theorem. Doing so, we get . Note that all the terms after are divisible by (the second term is just ), so we have is congruent to 1 mod , which means that is congruent to 0 mod , aka is divisible by .
RELATED QUESTIONS
Prove using math induction 2^n + (-1)^n+1 is divisible by... (answered by math_helper)

Using math of induction show that 2^n + (-1)^n+1 is divisible by... (answered by math_helper)

using mathematicsl induction prove that 3^n+7^n-2 is divisible by... (answered by Edwin McCravy)

Use mathematical induction to prove each statement is true for all positive integers n:... (answered by math_helper)

show that the number {{{n^(n-1)-1}}} is divisible by {{{(n-1)^2}}} whenever... (answered by Edwin McCravy)

Prove that the value of the expression is not divisible by 6 for any whole n:... (answered by MathLover1,greenestamps)

Let a_n = 11...1 with 3^n digits. Prove that a_n is divisible by... (answered by richard1234,robertb)

Find all positive integers n such that 𝑛 2 + 1 is divisible by n+... (answered by MathLover1)

Prove by mathematical induction that: 2^2n - 1 is divisible by 3 for all positive... (answered by Edwin McCravy)