Binomial Theorem
Sum of Binomial Coefficients via Integration
nta_pyq_2025_apr
Grade 11
Question:
If $\displaystyle\sum_{r=0}^{5} \frac{{}^{11}C_{2r+1}}{2r+2} = \frac{m}{n}$, $\gcd(m, n) = 1$, then $m - n$ is equal to ___
Step-by-Step Solution
Key Concept: Integrate $(1+x)^{11}$ over $[0,1]$ and $[-1,0]$, then subtract to isolate the sum of odd-indexed binomial coefficients divided by the corresponding indices.
Integrating $(1+x)^{11}$ over $[0,1]$: $\frac{2^{12}-1}{12} = C_0 + \frac{C_1}{2} + \cdots + \frac{C_{11}}{12}$. Integrating over $[-1,0]$: $\frac{1}{12} = C_0 - \frac{C_1}{2} + \frac{C_2}{3} - \cdots$. Subtracting: $\frac{2^{12}-2}{12} = 2\left(\frac{C_1}{2} + \frac{C_3}{4} + \cdots\right)$, giving $\sum = \frac{2^{11}-1}{12} = \frac{2047}{12}$. $m - n = 2047 - 12 = 2035$.
Correct Answer: 2035