jee-main 2024 Q83

jee-main · India · session2_08apr_shift1 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients

Let $\alpha = \sum _ { r = 0 } ^ { n } \left( 4 r ^ { 2 } + 2 r + 1 \right) ^ { n } C _ { r }$ and $\beta = \left( \sum _ { r = 0 } ^ { n } \frac { { } ^ { n } C _ { r } } { r + 1 } \right) + \frac { 1 } { n + 1 }$. If $140 < \frac { 2 \alpha } { \beta } < 281$, then the value of $n$ is $\_\_\_\_$

Let $\alpha = \sum _ { r = 0 } ^ { n } \left( 4 r ^ { 2 } + 2 r + 1 \right) ^ { n } C _ { r }$ and $\beta = \left( \sum _ { r = 0 } ^ { n } \frac { { } ^ { n } C _ { r } } { r + 1 } \right) + \frac { 1 } { n + 1 }$. If $140 < \frac { 2 \alpha } { \beta } < 281$, then the value of $n$ is $\_\_\_\_$

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