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Question: The formulae \(( a + b ) ^ { m } = a ^ { m } + m a ^ { m - 1 } b\) \(\frac{(2n)!}{n!n!}\) holds whe...

The formulae

(a+b)m=am+mam−1b( a + b ) ^ { m } = a ^ { m } + m a ^ { m - 1 } b (2n)!n!n!\frac{(2n)!}{n!n!} holds when.

A

C1C0+2C2C1+3C3C2+....+nCnCn−1=\frac{C_{1}}{C_{0}} + \frac{2C_{2}}{C_{1}} + \frac{3C_{3}}{C_{2}} + .... + \frac{nC_{n}}{C_{n - 1}} =

B

n(n−1)2\frac{n(n - 1)}{2}

C

n(n+2)2\frac{n(n + 2)}{2}

D

n(n+1)2\frac{n(n + 1)}{2}

Answer

n(n+1)2\frac{n(n + 1)}{2}

Explanation

Solution

The expression can be written as =5(5n)−4(4n)=5n+1−4n+1= 5(5^{n}) - 4(4^{n}) = 5^{n + 1} - 4^{n + 1}

Hence it is valid only when 1(1−x)(3−x)=12(11−x−13−x)\frac{1}{(1 - x)(3 - x)} = \frac{1}{2}\left( \frac{1}{1 - x} - \frac{1}{3 - x} \right).