Which of the following identities is always true for any sequences and and any…

MATHEMATICS · Single correct · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Which of the following identities is always true for any sequences (ak)(a_k) and (bk)(b_k) and any positive integer nn? (a) ∑k=1n(2ak−3bk)=(2−3)∑k=1n(ak+bk)\displaystyle \sum_{k=1}^{n}(2a_k-3b_k)=(2-3)\sum_{k=1}^{n}(a_k+b_k) (b) ∑k=1n(2ak−3bk)=(2n)∑k=1nak−(3n)∑k=1nbk\displaystyle \sum_{k=1}^{n}(2a_k-3b_k)=(2n)\sum_{k=1}^{n}a_k-(3n)\sum_{k=1}^{n}b_k (c) ∑k=1n(2ak−3bk)=2∑k=1nak−3∑k=1nbk\displaystyle \sum_{k=1}^{n}(2a_k-3b_k)=2\sum_{k=1}^{n}a_k-3\sum_{k=1}^{n}b_k (d) ∑k=1n(2ak−3bk)=2∑k=1nak+3∑k=1nbk\displaystyle \sum_{k=1}^{n}(2a_k-3b_k)=2\sum_{k=1}^{n}a_k+3\sum_{k=1}^{n}b_k

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