Answer:
[tex]5(\frac{1-(\frac{2}{3} )^8}{1-\frac{2}{3} })[/tex]
Step-by-step explanation:
The given series is [tex]\sum_{k=1}^85(\frac{2}{3})^{k-1}[/tex]
We want to write an expression that can be used to evaluate this series.
Since this is a geometric series, we use the formula for the sum of the first k terms.
[tex]S_k=a(\frac{1-r^k}{1-r})[/tex]
From the given series the first term is [tex]a=5[/tex] and the common ratio is [tex]r=\frac{2}{5}[/tex] and k=8
We substitute the values to obtain:
[tex]S_8=5(\frac{1-(\frac{2}{3} )^8}{1-\frac{2}{3} })[/tex]
