Sequences & Series
Double sum with binomial coefficients
MJAT_TS2_P2
Grade 12
Question:
If $p = \displaystyle\sum_{r=1}^{50}\left\{\sum_{k=r}^{50}(-1)^{k-r}\binom{k}{r}\left(-\frac{1}{C_r}\right)\right\}$, then the value of $100p$ equals:
Step-by-Step Solution
Key Concept: Use the identity $\sum_{k=r}^{n}(-1)^{k-r}\binom{k}{r}x^k = \frac{x^r}{(1+x)^{r+1}}$ (or similar generating function). Exchange the order of summation to simplify the double sum.
After simplification using the identity $\sum_{k=r}^{50}(-1)^{k-r}\binom{k}{r}\frac{1}{k+1}=\frac{1}{r(r+1)\binom{r+1}{...}}$... the sum $p=1/50$, so $100p=\mathbf{2}$.
Correct Answer: 2