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Fundamental Laws of Logarithms

Passage:If ...

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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 $a_{i}$ $r$ are
( $i = 0, 1, 2, 3, . . .2 n$ ) real numbers. On the basis of above information answer the following questions. The value of $2 n \sum r = 0 a_{2 r}$ 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 B $9^{n} + 1$
$(1 + 4 x + 2 x^{2})^{n}$
$(1 + 2 x)^{2 n}$ ------------(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)
Adding $a$ and $b$ , we get
$2 \sum_{0}^{n} a_{2 r} = 3^{2 n} + 1$
$= 9^{n} + 1$

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