IB Maths AA SL AA 1.3 practice question

Paper 1, no calculatorHard6 marks

For $n\in\mathbb Z$ define

S_n=(n+1)^5+(n-1)^5.

A student conjectures that $S_n$ is divisible by $32$ for every odd integer $n$

(a)

Calculate $S_1$ directly.

[1]

(b)

Starting from $S_n$ prove the identity

S_n\equiv 2n^5+20n^3+10n

for all $n\in\mathbb Z$ Use $=$ between successive expressions and $\equiv$ for the final identity.

[2]

(c)

Hence prove the student’s conjecture for every odd integer $n$

[3]