JEE Advanced: Nested Sum Collapses With One Elegant Idea!

Описание: JEE Advanced: Nested Sum Collapses With One Elegant Idea! | Factorial's Question of the Day

In this video from Factorial’s Question of the Day, we take a seemingly complicated nested sum with binomial coefficients and convert it into a beautiful integral expression that collapses the entire problem in just a few steps.
This is one of those problems that looks intimidating at first, but becomes incredibly elegant once you use the right identity:
If
p = ∑_{r=1}^{50} { ∑_{k=1}^{r} [ (-1)^{r-1} / k ] } · 50C_r,
then find the value of 200p.

This single observation transforms the entire inner sum into a closed form, and with one binomial identity, the complete expression melts into a simple integral.
Students will absolutely love this approach — it’s clean, powerful, and deeply satisfying.
📌 Concepts Used
Converting sums to integrals
Binomial expansion tricks
Swapping summations
Classic alternating sum identity
Beautiful collapse into a simple definite integral
🎯 Series: Factorial’s Question of the Day — JEE Advanced 2026
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