First slide

Binomial theorem for negative integral and rational Index

Question

$1 - \frac{1}{8} + \frac{1.3}{8.16} - \frac{1.3.5}{8.16.24} + .. =$

Moderate

A

$\sqrt{5}$
B

$1 / \sqrt{5}$
C

$2 / \sqrt{5}$
D

$\sqrt{5} / 2$

Solution

$\begin{array}{l} Comparing the given series with 1 + nx + \frac{n (n - 1)}{1.2} x^{2} + . . \\ We get nx = - \frac{1}{8}, \frac{n (n - 1)}{1.2} x^{2} = \frac{1.3}{8.16} \Rightarrow nx (nx - x) = \frac{3}{64} \\ \Rightarrow (- 1 / 8) (- 1 / 8 - x) = 3 / 64 \\ \Rightarrow 1 / 8 + x = 3 / 8 \Rightarrow x = 3 / 8 - 1 / 8 \\ = 2 / 8 = 1 / 4 \\ \Rightarrow nx = - 1 / 8 \Rightarrow n (1 / 4) = - 1 / 8 \Rightarrow n = - 1 / 2 \\ Given series = (1 + x)^{n} = (1 + 1 / 4)^{- 1 / 2} = (5 / 4)^{- 1 / 2} \\ = (4 / 5)^{1 / 2} = 2 / \sqrt{5} . \end{array}$

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