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Implicit Differentiation

Passage-1If ...

Question

Passage-1
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.
Sum of the coefficients is

A

4^{n}

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B

1

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C

9^{n}

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D

9

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Solution

The correct option is B $9^{n}$
$(1 + 4 x + 2 x^{2})^{n}$
$(1 + 2 x)^{2 n}$ ...(i)
$= \sum_{0}^{2 n} a_{r} x^{r}$
By substituting $x = 1$ in equation (i), we get the sum of the coefficients of the binomial expansion of $(1 + 2 x)^{2 n}$
Hence answer is
$(1 + 2)^{2 n}$
$= 3^{2 n}$
$= 9^{n}$

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