The formula for a finite geometric series is:
[tex]S_{n} = \frac{ a_{1}(1-r^{n}) }{1-r} [/tex]
a₁ is the first term of the geometric series. From the given series we can see that the first term of the series is 16.
r is the common ratio of the series. r can be found by dividing a term by its previous term. so r for the given series will be:
r = 24/16 = 3/2
n is the number of terms in the series. We can use the general formula of a Geometric Series to find the number of terms in the given series:
[tex]a_{n} = a_{1} (r)^{n-1} [/tex]
We want to find at which number in the series is 81 located. Using the values, we get:
[tex]81=16( \frac{3}{2} )^{n-1} \\ \\ \\
\frac{81}{16}= (\frac{3}{2})^{n-1} \\ \\
( \frac{3}{2} )^{4} =(\frac{3}{2})^{n-1} \\ \\
n=5 [/tex]
This means, there are 5 terms in the given series.
Using the values in the formula, we get:
[tex]S_{n}= \frac{16(1-( \frac{3}{2} )^{5}) }{1- \frac{3}{2} } \\ \\
S_{n}=211[/tex]
This means, the sum of given geometric series is 211.
