For each positive integer

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

For each positive integer nn, define (a) Pn=∑k=1n(3⋅2k−1−5)P_n=\displaystyle \sum_{k=1}^{n}\bigl(3\cdot2^{k-1}-5\bigr). Show that PnP_n can be written in the form A(2n−1)−5nA(2^{n}-1)-5n, where AA is a constant to be determined. (b) Using your result from part (a), compute P6P_{6}. (Present your answer as an integer.)

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