Mathematics

Problem

Define

S=\displaystyle \sum_{k=0}^{n} (-3)^{k}\,{n\choose k}\,2^{\,n-k}

. a. (i) Express

S

as the value of a single power expression obtained by expanding a suitable binomial. b. (ii) Hence evaluate

S

for (a)

n=5

and (b) general

n

. c. (iii) Using your result from part (ii), find the value of

\displaystyle \sum_{k=0}^{10} (-1)^{k}\,{10\choose k}\,3^{10-k}