Let . Answer the following questions based on the coefficients in this…

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

Problem

Let

P(x)=(1-5x)^{12}

. Answer the following questions based on the coefficients in this expansion. (i) Find the sum of all the coefficients of

P(x)

. (ii) Calculate the alternating sum of the coefficients of

P(x)

, that is,

\displaystyle\sum_{k=0}^{12}(-1)^k{12\brace k}(-5)^k

. (iii) Hence determine the sum of the coefficients of those terms in

P(x)

that contain an even power of

x

. Show all steps clearly.