If \((1/2^1) + (1/2^2) + (1/2^3)....(1/2^{10})=1/k\) , then what is the value of k?
1024/1023
This is a geometric series with \(a=\tfrac{1}{2}, r=\tfrac{1}{2}, n=10\).
\[S_{10}=\frac{\tfrac{1}{2}\bigl(1-(\tfrac{1}{2})^{10}\bigr)}{1-\tfrac{1}{2}}=1-\frac{1}{1024}=\frac{1023}{1024}\]
Setting \(\dfrac{1023}{1024}=\dfrac{1}{k}\) gives \(k=\dfrac{1024}{1023}\).
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