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  4. The sum of the series ( 101 C1/ 101 C0)+(2 ⋅ 101 C2/ 101 C1)+(3 ⋅ 101 C3/ 101 C2)+ ldots+(101 ⋅ 101 C101/ 101 C100) equals

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Q. The sum of the series
$\frac{{ }^{101} C_{1}}{{ }^{101} C_{0}}+\frac{2 \cdot{ }^{101} C_{2}}{{ }^{101} C_{1}}+\frac{3 \cdot{ }^{101} C_{3}}{{ }^{101} C_{2}}+\ldots+\frac{101 \cdot{ }^{101} C_{101}}{{ }^{101} C_{100}} $ equals

Binomial Theorem

Solution:

Consider the general term
$T_{r}=\frac{r \cdot{ }^{101} C_{r}}{{ }^{101} C_{r-1}} $ which simplifies to $(102-r)$
hence sum $=\displaystyle\sum_{r=1}^{101} 102-r$
$=101+100+99+\ldots+2+1=5151$