Mathematics

Problem

Consider the expression

\left(3-\dfrac{1}{2}y\right)^{8}

. a. (i) Rewrite it in the form

3^{8}\displaystyle\sum_{k=0}^{8} c_k\,y^{k}

and obtain a simplified expression for

c_k

. b. (ii) Using the expression for

c_k

, determine the coefficient of

y^{5}

. c. (iii) Prove that the ratio of the coefficients of the consecutive terms containing

y^{k}

and

y^{k+1}

\dfrac{8-k}{k+1}\,\cdot\dfrac{1}{6}