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Question

Find the value of $$C_{o}+\dfrac{C_{1}}{2}+\dfrac{C_{2}}{3}+......+\dfrac{C_{n}}{n+1}$$


Solution

We know that $$(1+x)^{n} = C_{0}+C_{1}x+C_{2}x^{2}+...c_{n}x^{n}$$ [Binomial Expansion ]
Integrating both sides,
$$ \int_{0}^{1}(1+x)^{n}dx = \int_{0}^{1}(C_{0}+C_{1}+C_{2}x^{2}+...C_{n}x^{n})dx$$
$$ \frac{(1+x)^{n}}{n+1} \int_{0}^{1} = C_{0}x+\frac{C_{1}x^{2}}{2}+\frac{C_{1}x^{3}}{3}+...\frac{C_{n}x^{n+1}}{n+1}\int_{0}^{1}$$
$$ \Rightarrow C_{0}+\frac{C_{1}}{2}+\frac{C_{2}}{3}+\frac{C_{n}}{n+1} = \frac{2^{n}}{n+1}$$

1118882_1202301_ans_aa09c8dacb0048ddacd978837d0ff225.jpg

Mathematics

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