First slide

Binomial theorem for negative integral and rational Index

Question

$The coefficient of x^{2} in \frac{1 + x^{2}}{(1 - x)^{3}} is$

Moderate

A

3
B

4
C

7
D

2

Solution

$\begin{array}{l} \frac{1 + x^{2}}{(1 - x)^{3}} = (1 + x^{2}) (1 - x)^{- 3} \\ = (1 + x^{2}) [1 +^{3} C_{1} x +^{4} C_{2} x^{2} +^{5} C_{3} x^{3} + \dots] \\ = (1 + x^{2}) (1 + 3 x + 6 x^{2} + \dots) \\ Coefficient of x^{2} = 1 + 6 = 7 . \end{array}$

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