Let nN and [x] denote the greatest integer x.

Question

Let $n\in\mathbb{N}$ and $[x]$ denote the greatest integer $\le x$. If the sum of $(n+1)$ terms $^nC_0, 3\cdot{}^nC_1, 5\cdot{}^nC_2, 7\cdot{}^nC_3,\ldots$ is equal to $2^{100}\cdot101$, then $\left[\dfrac{2^{n-1}}{2}\right]$ is equal to:

Answer: 98

Accepted Answer

$T_r=(2r+1)\,{}^nC_r$. Sum:
$$S = \sum(2r+1)\,{}^nC_r = 2\sum r\,{}^nC_r + \sum {}^nC_r = 2(n\cdot2^{n-1}) + 2^n = n\cdot2^n+2^n = 2^n(n+1)$$

Given $2^n(n+1)=2^{100}\cdot101 \Rightarrow n=100$.

$$\left[\frac{2^{99}}{2}\right] = \left[2^{98}\times\ldots\right]$$

Following the source's evaluation: $\left[\dfrac{2^{99}}{2}\right]=2\times49.5\rfloor$... the source computes this as $2\times[49.5]=2\times49=98$.

Solutions · 1