Question:
The coefficient of $x^{50}$ in the expansion of $\frac{\left(3-5x\right)}{\left(1-x\right)^{2}}$ equals
The coefficient of $x^{50}$ in the expansion of $\frac{\left(3-5x\right)}{\left(1-x\right)^{2}}$ equals
Updated On: Jul 7, 2022
- $99$
- $403$
- $-97$
- $51$
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The Correct Option is C
Solution and Explanation
$\frac{3-5x}{\left(1-x\right)^{2}} = \left(3-5x\right) \left(1-x\right)^{-2}$
$= \left(3 - 5x\right) \left(1 + 2x + 3x^{2 }+ ... + 50x^{49} + 51x^{50} +...\right)$
$\therefore$ Required coefficient $= 3 \times 5 1 - 5 \times 50$
$= 153 - 250 = - 97$
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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 .