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Index of (r+1)th Term from End When Counted from Beginning

If n be a pos...

Question

If n be a positive integer, then in the trinomial expansion of $(x^{2} + 2 x + 2)^{n}$ the coefficient of

$x i s 2^{n} . n$

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$x i s 2^{n}$

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$x^{2} i s n^{2} {.2}^{n - 1}$

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$x^{3} i s 2^{n}$

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Solution

The correct options are
A
$x i s 2^{n} . n$

C
$x^{2} i s n^{2} {.2}^{n - 1}$

$(x^{2} + 2 x + 2)^{n} = (2 + x (2 + x))^{n}$

General term of the expansion $= \sum_{r = 0}^{n} {^{n} C}_{r} {.2}^{n - r} x^{r} (2 + x)^{r}$

Coefficient of $x = 2^{n} . n$

Coefficient of $x^{2} = 2^{n - 1} . n + \frac{n (n - 1) 2^{n - 2} 2^{2}}{2} = n^{2} {.2}^{n - 1}$

Coefficient of $x^{3} =^{n} C_{2} {.2}^{n - 2} .4 +^{n} C_{3} 2^{n - 3} {.2}^{3}$

$= 2^{n} .^{n + 1} C_{3}$

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