JEE Mains · Maths · STD 11 - 8. sequence and series
The value of \( \frac{^{100}C_{50}}{51} + \frac{^{100}C_{51}}{52} + \dots + \frac{^{100}C_{100}}{101} \) is:
- A \( \frac{2^{101}}{100} \)
- B \( \frac{2^{100}}{100} \)
- C \( \frac{2^{101}}{101} \)
- D \( \frac{2^{100}}{101} \)
Answer & Solution
Correct Answer
(D) \( \frac{2^{100}}{101} \)
Step-by-step Solution
Detailed explanation
\(S=\sum_{r=50}^{100} \frac{{ }^{100} C_r}{r+1}=\sum_{r=50}^{100} \frac{1}{r+1} \cdot \frac{r+1}{101} \cdot{ }^{101} C_{r+1}\) \(S =\frac{1}{101} \sum_{ r =50}^{100}{ }^{101} C _{ r +1}\) \(=\frac{1}{101} \times \frac{2^{101}}{2}=\frac{2^{100}}{101}\)
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