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Proof by mathematical induction — Question 3

QCAA 2021, Paper 2 · 1 mark

Q3 · 2021 · Technology-activeSimple familiar1 mark

QUESTION 3

Given n∈Z+n\in\mathbb Z^+, for which proposition can the initial statement for mathematical induction be proven?
(A)
x2n−y2nx^{2n}-y^{2n} is divisible by (x+y)(x+y) ∀(x+y)≠0\forall(x+y)\ne0
(B)
12+22+32+⋯+n2=16n(2n2+3n−1)1^2+2^2+3^2+\cdots+n^2=\frac16n(2n^2+3n-1)
(C)
(n+1)3+(n+2)3(n+1)^3+(n+2)^3 is divisible by 33
(D)
∑r=1n1(2r−1)(2r+1)=nn+1\displaystyle \sum_{r=1}^{n}\frac{1}{(2r-1)(2r+1)}=\frac{n}{n+1}
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