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Sum of Coefficients of All Terms

PassageIf n...

Question

Passage
If $n$ is the positive integer and if $(1 + 4 x + 4 x^{2})^{n} = 2 n \sum r = 0 a_{r} x^{r}$ , where $a_{r}$ are real numbers. On the basis of above information answer the following questions. The value of $2 n \sum r = 1 a_{2 r - 1}$ is

A

9^{n} - 1

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B

9^{n} + 1

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C

9^{n} - 2

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D

9^{n} + 2

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Solution

The correct option is A $9^{n} - 1$
$(1 + 4 x + 2 x^{2})^{n}$
$(1 + 2 x)^{2 n}$ ...(i)
$= \sum_{0}^{2 n} a_{r} x^{r}$
Substituting $x = 1$ in equation (i), we get
$3^{2 n}$ ...(a)
Substituting $x = - 1$ in equation (i), we get
$(- 1)^{2 n}$ ...(b)
Subtracting $b$ from $a$ , we get
$2 \sum_{0}^{n} a_{2 r - 1} = 3^{2 n} - 1$
$= 9^{n} + 1$

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