Question

Let \(n\in N\) and \([x]\) denote the greatest integer less than or equal to \(x\). If the sum of \((n+1)\) terms \({}^nC_0,\ 3\cdot{}^nC_1,\ 5\cdot{}^nC_2,\ 7\cdot{}^nC_3,\ \dots\) is equal to \(2^{100}\cdot101\), then \(2\left[\frac{n-1}{2}\right]\) is equal to _______________.

Hint:

The sum is \(\sum(2r+1)\binom nr=(n+1)2^n\).

Answer 1

Correct Answer: 98

Sum \(=\sum_{r=0}^n(2r+1)\binom nr=2\sum r\binom nr+\sum\binom nr=2\cdot n2^{n-1}+2^n=(n+1)2^n\).
\((n+1)2^n=101\cdot2^{100}\Rightarrow n=100\).
\(2\left[\frac{99}{2}\right]=2\times49=98\).
Answer: 98