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Sagot :
We are given the following geometric series
[tex]64+96+144+216+...[/tex]Common ratio:
The common ratio (r) of a geometric series can be found as
r = 96/64 = 1.5
you can take any two consecutive terms, the common ratio will always be the same
r = 144/96 = 1.5
r = 216/144 = 1.5
Therefore, the common ratio of the given geometric series is 1.5
Now let us find the sum of the first 10 terms of this series.
The sum of a geometric series is given by
[tex]S_n=\frac{a_1(1-r^n)}{1-r}[/tex]Where a₁ is the first term of the series, r is the common ratio, and n is the number of terms.
For the given case
a₁ = 64
r = 1.5
n = 10
Let us substitute these values into the above formula
[tex]\begin{gathered} S_{10}=\frac{64\cdot_{}(1-1.5^{10})}{1-1.5} \\ S_{10}=\frac{64\cdot_{}(1-57.665^{})}{-0.5} \\ S_{10}=\frac{64\cdot_{}(-56.665^{})}{-0.5} \\ S_{10}=\frac{-3626.56}{-0.5} \\ S_{10}=7253.12 \end{gathered}[/tex]Therefore, the sum of the first 10 terms of this series is 7253.12
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