First slide

Binomial theorem for positive integral Index

Question

Let $C_{r} =^{15} C_{r}, 0 \leq r \leq 15 .$ Sum of the series $S = \sum_{r = 1}^{15} r (\frac{C_{r}}{C_{r - 1}})$ is

Moderate

A

40
B

60
C

100
D

120

Solution

$\begin{array}{l} \frac{C_{r}}{C_{r - 1}} = \frac{15!}{r! (15 - r)!} \times \frac{(r - 1)! (16 - r)!}{15!} \\ = \frac{16 - r}{r} \\ \Rightarrow S = \sum_{r = 1}^{15} r (\frac{C_{r}}{C_{r - 1}}) = \sum_{r = 1}^{15} (16 - r) \\ = \frac{1}{2} (15) (16) = 120 \end{array}$

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