Find the coefficient of - a12 in the expansion of 4-2a-a21-a3

⟹( 4 2a a ^ 2 ) ( 1+a ) #160; ^ 3 ( 1+a ) #160; ^ 3 : T _ r+1 =^ n+r 1 C _ r ( 1 ) #160; ^ r · a ^ r For #160; a ^ 12 Powers o ...

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Standard XII

Mathematics

Combination

Find the coef...

Question

Find the coefficient of :
$a^{12}$ in the expansion of $\frac{4 + 2 a - a^{2}}{(1 + a)^{3}}$

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Solution

$⟹ (4 - 2 a - a^{2}) {(1 + a)}^{- 3}$
${(1 + a)}^{- 3} : -$
$T_{r + 1} =^{n + r - 1} C_{r} {(- 1)}^{r} \cdot a^{r}$
For $a^{12}$
Powers of $^{'} a^{'}$ needed is $: - 12, 11, 10$
$T_{13} =^{3 + 12 - 1} C_{12} {(- 1)}^{12} =^{14} C_{12}$
$T_{12} =^{3 + 11 - 1} C_{11} {(- 1)}^{11} = -^{13} C_{11}$
$T_{11} =^{12} C_{10}$
$∴$ Coefficient of $a^{12}$ in above expression is :-
$⟹ 4 \times^{14} C_{12} - 2 \times^{13} C_{11} -^{12} C_{10}$
$⟹ 142$

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Find the coefficient of :a12 in the expansion of 4+2a−a2(1+a)3

⟹(4−2a−a2)(1+a)−3(1+a)−3:−Tr+1=n+r−1Cr(−1)r⋅arFor a12Powers of ′a′ needed is :−12,11,10T13=3+12−1C12(−1)12=14C12T12=3+11−1C11(−1)11=−13C11T11=12C10∴ Coefficient of a12 in above expression is :-⟹4×14C12−2×13C11−12C10⟹142

Find the coefficient of :
$a^{12}$ in the expansion of $\frac{4 + 2 a - a^{2}}{(1 + a)^{3}}$