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The coefficient of ${x^n}$ in the expression ${{5x + 6} \over {(2 + x)\,(1 - x)}}$ when expanded in ascending order is
${{ - \,2} \over 3}\,{{{{( - 1)}^n}} \over {{2^n}}} + {{11} \over 3}$
${2 \over 3} + {{{{( - 1)}^n}} \over {{2^n}}} - {{11} \over 3}$
$ - {2 \over 3} + {{{{( - 1)}^n}} \over 3} - {{11} \over {{2^n}}}$
None of these
Solution
(a) ${{5x + 6} \over {(2 + x)\,(1 – x)}} = {{{{ – 4} \over 3}} \over {2 + x}} + {{{{11} \over 3}} \over {1 – x}}$
Rewriting the denominators for expressions, we get
= ${{{{ – 4} \over 3}} \over {2\left( {1 + {x \over 2}} \right)}} + {{{{11} \over 3}} \over {1 – x}} = {{ – 2} \over 3}{\left( {1 + {x \over 2}} \right)^{ – 1}} + {{11} \over 3}{(1 – x)^{ – 1}}$
= ${{ – 2} \over 3}\left[ {1 – {x \over 2} + {{{x^2}} \over 4} – {{{x^3}} \over 8} + …… + {{( – 1)}^n}{{{x^n}} \over {{2^n}}} + ……} \right]\,$
$ + {{11} \over 3}[1 + x + {x^2} + ……. + {x^n} + …..]$
The coefficient of ${x^n}$ in the given expression is
${{ – 2} \over 3}{( – 1)^n}{1 \over {{2^n}}} + {{11} \over 3}$.
