Mathematics

Problem

For each positive integer

n

, define a. (a)

P_n=\displaystyle \sum_{k=1}^{n}\bigl(3\cdot2^{k-1}-5\bigr)

. Show that

P_n

can be written in the form

A(2^{n}-1)-5n

, where

A

is a constant to be determined. b. (b) Using your result from part (a), compute

P_{6}

. (Present your answer as an integer.)