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JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Binomial Theorem

Suppose

\sum_{r = 0}^{2023} r^{2}^{2023} C_{r} = 2023 \times α \times 2^{2022}

. Then the value of

α

is

Answer: 1012

Solution

Using the identity

r = 0 \sum n r^{2} (r n) = n (n + 1) 2^{n - 2}

with

n = 2023

, the sum is

2023 \cdot 2024 \cdot 2^{2021}

. Writing

2024 = 1012 \cdot 2

turns this into

2023 \cdot 1012 \cdot 2^{2022}

. Comparing with

2023 \times α \times 2^{2022}

, the value is

α = 1012.

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Integer answer

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