Find the sum of the first six terms in the geometric sequence \(\frac12,\frac14,\frac18,\dots\). Express your answer as a common fraction.
S = (1/2) ( 1 - (1/2)^6 ) / ( 1 - 1/2) = 63/64 ( we could see this from the other problem )
This 6-term geometric series has first term \(a_0 = \frac12\) and ratio 1/2, so it has value
\(\begin{align*} \frac{\frac12(1-\left(\frac12\right)^{6})}{1-\frac12} &= 1-\left(\frac12\right)^{6}\\ &= 1-\frac1{64}\\ &= \boxed{\frac{63}{64}}. \end{align*}\)
+285 
