Question

The coefficient of $x^{n}$ in the expression $\frac{5x + 6}{(2 + x)(1 - x)}$ when expanded in ascending order is

• A. $\frac{- 2}{3}\frac{( - 1)^{n}}{2^{n}} + \frac{11}{3}$
• B. $\frac{2}{3} + \frac{( - 1)^{n}}{2^{n}} - \frac{11}{3}$
• C. $- \frac{2}{3} + \frac{( - 1)^{n}}{3} - \frac{11}{2^{n}}$
• D. None of these

Answer 1

Answer: (A)

$\frac{- 2}{3}\frac{( - 1)^{n}}{2^{n}} + \frac{11}{3}$

Answer 2

$\frac{5x + 6}{(2 + x)(1 + x)} = \frac{\frac{- 4}{3}}{2 + x} + \frac{\frac{11}{3}}{1 - x}$Rewriting the denominators for expressions, we get

=$\frac{\frac{- 4}{3}}{2\left( 1 + \frac{x}{2} \right)} + \frac{\frac{11}{3}}{1 - x} = \frac{- 2}{3}\left( 1 + \frac{x}{2} \right)^{- 1} + \frac{11}{3}(1 - x)^{- 1}$

= $\frac{- 2}{3}\left\lbrack 1 - \frac{x}{2} + \frac{x^{2}}{4} - \frac{x^{3}}{8} + ...... + ( - 1)^{n}\frac{x^{n}}{2^{n}} + ...... \right\rbrack + \frac{11}{3}\lbrack 1 + x + x^{2} + ....... + x^{n} + .....\rbrack$The coefficient of $x^{n}$ in the given expression is

$\frac{- 2}{3}( - 1)^{n}\frac{1}{2^{n}} + \frac{11}{3}$.