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

Question

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 (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Consider the general term Tr=(r ⋅ 101 Cr/ 101 Cr-1) which simplifies to (102-r) hence sum = displaystyle∑r=1101 102-r =101+100+99+ ldots+2+1=5151

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