Answer:
The equation is:
[tex]S=\frac{a_n*r-a_1}{r-1}[/tex]
[tex]S=\frac{\frac{16}{243}*\frac{2}{3}-\frac{1}{3}}{\frac{2}{3}-1}[/tex]
The sum is:
[tex]S=\frac{211}{243}[/tex]
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If the sequence is infinite, the formula is:
[tex]S = \frac{a_1}{1-r}[/tex]
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Step-by-step explanation:
We must calculate the radius of the geometric series
[tex]r =\frac{a_{n+1}}{a_n}\\\\r=\frac{\frac{2}{9}}{\frac{1}{3}}\\\\r=\frac{2}{3}[/tex]
The first term of the series is: [tex]a_1=\frac{1}{3}[/tex]
The last term of the series is: [tex]a_n=\frac{16}{243}[/tex]
If the sequence is finite then the formula is:
[tex]S=\frac{a_n*r-a_1}{r-1}[/tex]
[tex]S=\frac{\frac{16}{243}*\frac{2}{3}-\frac{1}{3}}{\frac{2}{3}-1}[/tex]
[tex]S=\frac{211}{243}[/tex]
If the sequence is infinite then by definition as the radius are [tex]0 <| r | <1[/tex] then the formula for the sum of the geometric sequence is:
[tex]S = \frac{a_1}{1-r}\\\\S = \frac{\frac{1}{3}}{1-\frac{2}{3}}\\\\S =1[/tex]
