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Sum of Coefficients of All Terms

If n is a p...

Question

If $n$ is a positive integer, then the coefficient of $x^{n}$ in the expansion of $\frac{(1 + 2 x)^{n}}{1 - x}$ is

n {.3}^{n}

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(n - 1) 3^{n}

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(n + 1) 3^{n}

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3^{n}

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Solution

The correct option is B $3^{n}$
To find the coefficient of $x^{n}$ in the expansion of $\frac{(1 + x)^{2 n}}{1 - x}$
This can be written as $(1 + x)^{2 n} (1 - x)^{- 1}$
$= ((\frac{2 n}{0}) + (\frac{2 n}{1}) x + (\frac{2 n}{2}) x^{2} + (\frac{2 n}{3}) x^{3} + . . . . . . . . . . . . + (\frac{2 n}{n}) x^{n}) (1 + x + x^{2} + x^{3} + x^{4} + . . .)$
On Multiplying this, we can find the coefficient of $x^{n}$ as summation of
$= (\frac{2 n}{0}) + (\frac{2 n}{1}) + (\frac{2 n}{2}) . . . . . . . + (\frac{2 n}{n})$
$= 3^{n}$

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(1 + x)^{2} + (1 + x)^{3} + \dots + (1 + x)^{49} + (1 + m x)^{50} i s (3 n + 1)^{51} C_{3}

x^{- 1}

(1 + x)^{n} (1 + \frac{1}{x})^{n}

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n

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