If then k is equal to

JEE MainMathematicsToughBinomial Theorem

If
$a_{n} = \sum_{r = 0}^{n}\mspace{2mu}\frac{1}{\ ^{n}C_{r}},\ then\ \sum_{r = 0}^{n}\mspace{2mu}\frac{r}{\ ^{n}C_{r}} = \frac{k}{4}n \cdot a_{n}$
then k is equal to

Show answer and solution

We have

S = \sum_{r = 0}^{n}\mspace{2mu}\frac{r}{\ ^{n}C_{r}} = \sum_{r = 0}^{n}\mspace{2mu}\frac{n - r}{\ ^{n}C_{n - r}} = \sum_{r = 0}^{n}\mspace{2mu}\frac{n}{\ ^{n}C_{r}} - \sum_{r = 0}^{n}\mspace{2mu}\frac{r}{\ ^{n}C_{r}}

= na_{n} - S

\Rightarrow S = \frac{1}{2}na_{n} = \frac{2}{4}na_{n} \Rightarrow k = 2

Source: Grass-G / M-28 Mathematics Paper FST 16 Jan.docx

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