The summation of the given geometric series is 70.5.
Geometric series is the sum of an infinite number of terms that have a constant ratio between the successive terms.
[tex]s_{n}= a\frac{(r^{n} -1)}{(r-1)}[/tex] (when r>1)
or
[tex]S_{n} =\frac{a(r^{n}-1) }{1-r}[/tex] (when r<1)
where,
r is the common ratio
a is the first term of first term of the series.
n is the total number of terms.
According to the given question
we have
geometric series [tex]\frac{1}{2} +2+8+32+128[/tex]
total number of terms, n=5
first term a =[tex]\frac{1}{2}[/tex]
[tex]r=\frac{2}{\frac{1}{2} }=4[/tex]
therefore,
summation of the given geometric series calculated as
[tex]s_{n} =\frac{1}{2} \frac{4^{5}-1 }{4-1}[/tex]
[tex]s_{n} =\frac{1023}{6}[/tex]
[tex]s_{n} = 170.5[/tex]
Hence, the summation of the geometric series is 170.5.
Learn more about the summation of geometric series here:
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