AP EAMCET · Maths · Binomial Theorem
If \(|x| < \frac{1}{2}\), then the coefficient of \(x^r\) in the expansion of \(\frac{1+2 x}{(1-2 x)^2}\), is
- A \(r 2^r\)
- B \((2 r-1) 2^r\)
- C \(r 2^{2 r+1}\)
- D \((2 r+1) 2^r\)
Answer & Solution
Correct Answer
(D) \((2 r+1) 2^r\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \frac{1+2 x}{(1-2 x)^2}=(1+2 x)(1-2 x)^{-2} \\ &=(1+2 x)\left(1+\frac{2}{1 !}(2 x)+\frac{23}{2 !}+\ldots+\frac{23 \ldots r}{(r-1) !}(2 x)^{r-1}\right. \\ &\left.+\frac{2 \cdot 3 \cdot 4 \ldots(r+1)(2 x)^r}{r !}\right) \end{aligned}\) The coefficient of…
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