Suppose r = 0 2023 r 2 C r 2023 = 2023 × × 2 2022 . Then the value of is ___________ . [2023]

Question

Suppose $\sum_{r = 0}^{2023} r^{2} {}^{2023}{C_{r}} = 2023 \times α \times 2^{2022}$ . Then the value of $α$ is ___________ . [2023]

Smart Achievers · Accepted Answer

(1012)

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

$\sum_{r = 0}^{n} r^{2} \cdot {}^{n}{C_{r}} = n (n + 1) \cdot 2^{n - 2}$

Then, $\sum_{r = 0}^{2023} r^{2} \cdot {}^{2023}{C_{r}} = (2023) (2023 + 1) \cdot 2^{2023 - 2}$

$= 2023 \times 2024 \times 2^{2021} = 2023 \times α \times 2^{2022}$

$∴$ $2 α = 2024 \Rightarrow α = 1012$

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