Question

What is the sum of the first eight terms of a geometric series whose first term is 3 and whose common ratio is .5 ?

Answer 1

`\bf \qquad \textit{sum of a finite geometric sequence} \\ \quad \\ S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases} n=\textit{last term}\\ a_1=\textit{first term}\\ r=\textit{common ratio} \end{cases} \\ \quad \\ S_8=3\left( \cfrac{1-0.5^8}{1-0.5} \right)`

Answer 2

Answer: The required sum of first eight terms of the given geometric series is 5.98.

Step-by-step explanation: We are given to find the sum of first eight terms of a geometric series whose first term is 3 and whose common ratio is 0.5.

We know that

the sum of first n terms of a geometric series with first term a and common ratio r is given by

`S_n=(a(1-r^n))/(1-).`

For the given geometric series, we have

first term, a = 3 and common ratio, r = 0.5.

So, the sum of first eight terms of the given geometric series will be

`S_8\\\\\\=(a(1-r^8))/(1-r)\\\\\\=(3\left(1-\left((1)/(2)\right)^8\right))/(1-(1)/(2))\\\\\\=(3\left(1-(1)/(256)\right))/((1)/(2))\\\\\\=3*2*(256-1)/(256)\\\\\\=3*(255)/(128)\\\\\\=(765)/(128)\\\\=5.98.`

Thus, the required sum of first eight terms of the given geometric series is 5.98.