Let n be a positive integer.

Question

Let $n$ be a positive integer. Let

$$A=\sum_{k=0}^{n}(-1)^k\,{}^nC_k\left[\left(\frac12\right)^k+\left(\frac34\right)^k+\left(\frac78\right)^k+\left(\frac{15}{16}\right)^k+\left(\frac{31}{32}\right)^k\right]$$

If $63A=1-\dfrac{1}{2^{30}}$, then $n$ is equal to

Answer: 6

Accepted Answer

Using $(1-x)^n=\displaystyle\sum_{r=0}^n(-1)^r{}^nC_rx^r$ at $x=\dfrac12,\dfrac34,\dfrac78,\dfrac{15}{16},\dfrac{31}{32}$ (i.e. $x=1-\dfrac1{2^j}$ for $j=1,\dots,5$):

$$A=\sum_{j=1}^5\left(1-\left(1-\frac{1}{2^j}\right)\right)^n\cdot(-1)^{?}\;\Rightarrow\; A=\sum_{j=1}^{5}\frac{1}{2^{jn}}=\frac{1}{2^n}\cdot\frac{1-\left(\frac{1}{2^n}\right)^5}{1-\frac{1}{2^n}}=\frac{2^{5n}-1}{2^n-1}\cdot\frac{1}{2^{5n}}$$

Given $63A=1-\dfrac{1}{2^{30}}=\dfrac{2^{30}-1}{2^{30}}$; testing $n=6$ satisfies the equation exactly.

$n=6$

Solutions · 1