Coefficient of $x^{n - 1}$ in the expansion of. $(x + 3)^{n} + (x + 3)^{n - 1} (x + 2) + (x + 3)^{n - 2} (x + 2)^{2} + . . . . . . . + (x + 2)^{n}$ is

^{n + 1} C_{2} (3)

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^{n - 1} C_{2} (5)

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^{n + 1} C_{2} (5)

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^{n} C_{2} (5)

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Solution

The correct option is C $^{n + 1} C_{2} (5)$
The Series given is Geometric series with Common ratio $r = \frac{(x + 2)}{(x + 3)}$
Thus the sum of series is $S = (x + 3)^{n} \frac{1 - \frac{(x + 2)^{n + 1}}{(x + 3)^{n + 1}}}{1 - \frac{x + 2}{x + 3}}$
$S = \frac{(x + 3)^{n + 1} - (x + 2)^{n + 1}}{x + 3 - (x + 2)}$
$S = (x + 3)^{n + 1} - (x + 2)^{n + 1}$
Now the coefficient of $x^{n - 1}$ will be
$^{n + 1} C_{2} x^{n - 1} 3^{2} -^{n + 1} C_{2} x^{n - 1} 2^{2}$
$\Rightarrow^{n + 1} C_{2} (3^{2} - 2^{2})$
$\Rightarrow^{n + 1} C_{2} (5)$

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Similar questions

x^{r} [0 \leq r \leq (n - 1)]

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(x + 3)^{n - 1} + (x + 3)^{n - 2} (x + 2) + (x + 3)^{n - 3} (x + 2)^{2} + \dots + (x + 2)^{n - 1}

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