Binomial Theorem
Sum with Binomial Coefficients
nta_pyq_2024_apr
Grade None
Question:
Let $\alpha=\displaystyle\sum_{r=0}^{n}(4r^2+2r+1)\binom{n}{r}$ and $\beta=\left(\displaystyle\sum_{r=0}^{n}\dfrac{\binom{n}{r}}{r+1}\right)+\dfrac{1}{n+1}$. If $140<\dfrac{2\alpha}{\beta}<281$, then the value of $n$ is _______.
Step-by-Step Solution
Key Concept: $2\alpha/\beta=(n+1)^3$. $140<(n+1)^3<281\Rightarrow n+1=6\Rightarrow n=5$.
$n=5$.
Correct Answer: 5