+285 Find the sum of the first five terms in the geometric sequence \(\frac13,\frac19,\frac1{27},\dots\). Express your answer as a common fraction.
+129852 Sum of a geometric series
S = first term ( 1 - common ratio ^n) / ( 1 - common ratio) where n = the number of terms we are summing
The common ratio is (1/9) / (1/3) = 3/9 = 1/3
So
S = ( 1/3) ( 1 - (1/3)^3 ) / ( 1 - 1/3) =
(1/3) ( 1 - 1/27) / ( 2/3) =
(1/3) (3/2) * ( 26/27 ) =
(1/2) ( 26/27) =
26/54 =
13/27
+285 CPhill I got something different
This is a finite geometric series with first term \(1/3\) and common ratio \(1/3\). There are five terms, so the sum of this series is \(\frac{\frac13\left(1-\left(\frac13\right)^5\right)}{1-\frac13} = \boxed{\frac{121}{243}} \).
