Question:
The coefficient of $x^n$ in the expansion of
($1 - 2x + 3x^{2} - 4x^{3} + ...$ to $\infty$)$^{-n}$ is
The coefficient of $x^n$ in the expansion of
($1 - 2x + 3x^{2} - 4x^{3} + ...$ to $\infty$)$^{-n}$ is
Updated On: Jul 7, 2022
- $\frac{\left(2n\right)!}{n!\left(n-1\right)!}$
- $\frac{\left(2n\right)!}{\left[\left(n-1\right)!\right]^{2}}$
- $\frac{\left(2n\right)!}{\left(n!\right)^{2}}$
- None of these
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The Correct Option is C
Solution and Explanation
We have, ($1 - 2x + 3x^{2} - 4x^{3} + ...$ to $\infty$)$^{-n}$
$= \left[\left(1+x\right)^{-2}\right]^{-n} = \left(1+x\right)^{2n}$
$\therefore$ Coefficient of $x^{n} =\,^{2n}C_{n} = \frac{\left(2n\right)!}{\left(n!\right)^{2}}$.
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Notes on binomial expansion formula
Concepts Used:
Binomial Expansion Formula
The binomial expansion formula involves binomial coefficients which are of the form
(n/k)(or) nCk and it is calculated using the formula, nCk =n! / [(n - k)! k!]. The binomial expansion formula is also known as the binomial theorem. Here are the binomial expansion formulas.
This binomial expansion formula gives the expansion of (x + y)n where 'n' is a natural number. The expansion of (x + y)n has (n + 1) terms. This formula says:
We have (x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn
General Term = Tr+1 = nCr xn-r . yr
- General Term in (1 + x)n is nCr xr
- In the binomial expansion of (x + y)n , the rth term from end is (n – r + 2)th .