If 1n+1^nC_n+1n^nC_n-1++12^nC_1+^nC_0=102310, then n is equal to

Question

If $\dfrac{1}{n+1}{}^nC_n+\dfrac1n{}^nC_{n-1}+\cdots+\dfrac12{}^nC_1+{}^nC_0=\dfrac{1023}{10}$, then $n$ is equal to

Accepted Answer

$$\sum_{r=0}^n\frac{{}^nC_r}{r+1}=\frac{1}{n+1}\sum_{r=0}^n{}^{n+1}C_{r+1}=\frac{1}{n+1}(2^{n+1}-1)=\frac{1023}{10}$$

$$n+1=10\quad\Rightarrow\quad n=9$$

Solutions · 1