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Question: The value of \(\frac{C_{0}}{1.3} - \frac{C_{1}}{2.3} + \frac{C_{2}}{3.3} - \frac{C_{3}}{4.3} + .... ...

The value of C01.3−C12.3+C23.3−C34.3+....+(−1)nCn(n+1).3\frac{C_{0}}{1.3} - \frac{C_{1}}{2.3} + \frac{C_{2}}{3.3} - \frac{C_{3}}{4.3} + .... + ( - 1)^{n}\frac{C_{n}}{(n + 1).3} is

A

3n+1\frac{3}{n + 1}

B

n+13\frac{n + 1}{3}

C

13(n+1)\frac{1}{3(n + 1)}

D

None

Answer

13(n+1)\frac{1}{3(n + 1)}

Explanation

Solution

(1 + x)n = C0 + C1x +C2x2 +……..+ Cnxn

On integrating from – 1 to 0

C0 – C12+C23\frac{C_{1}}{2} + \frac{C_{2}}{3}……..+ (–1)nCnn+1=1n+1\frac{C_{n}}{n + 1} = \frac{1}{n + 1}

Hence required = 13(n+1)\frac{1}{3(n + 1)}