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Proof by mathematical induction — Question 8

QCAA 2021, Paper 1 · 1 mark

Q8 · 2021 · Technology-freeSimple familiar1 mark

QUESTION 8

Let P(n)P(n) be the proposition that
∑r=1n(r+1)3r−1=n×3n∀n∈Z+\displaystyle \sum_{r=1}^{n}(r+1)3^{r-1}=n\times3^n\quad\forall n\in\mathbb Z^+.
Which option represents a correct formulation of the assumption that P(k)P(k) is true ∀k∈Z+\forall k\in\mathbb Z^+ in a proof using mathematical induction?
(A)
∑r=1k(k+1)3k−1=k×3k\displaystyle \sum_{r=1}^{k}(k+1)3^{k-1}=k\times3^k
(B)
∑r=1k(k+1)3k−1=n×3n\displaystyle \sum_{r=1}^{k}(k+1)3^{k-1}=n\times3^n
(C)
∑r=1k(r+1)3r−1=k×3k\displaystyle \sum_{r=1}^{k}(r+1)3^{r-1}=k\times3^k
(D)
∑r=1k(r+1)3r−1=r×3r\displaystyle \sum_{r=1}^{k}(r+1)3^{r-1}=r\times3^r
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