Binomial Theorem
Telescoping Binomial Sum
nta_pyq_2023_apr
Grade 11
Question:
If $\dfrac{1}{n+1}\binom{n}{n}+\dfrac{1}{n}\binom{n}{n-1}+\cdots+\dfrac{1}{2}\binom{n}{1}+\binom{n}{0}=\dfrac{1023}{10}$, then $n$ is equal to
Step-by-Step Solution
Key Concept: $\frac{1}{r+1}\binom{n}{r}=\frac{1}{n+1}\binom{n+1}{r+1}$. So the sum $=\frac{1}{n+1}\sum_{r=0}^n\binom{n+1}{r+1}=\frac{2^{n+1}-1}{n+1}$.
$\frac{2^{n+1}-1}{n+1}=\frac{1023}{10}\Rightarrow n=9$.
Correct Answer: 1