Question

Find the sum of the following geometric series:

8 + 1.6 + 0.32 + 0.064 + ...

Answer 1

10

Explanation:

The given series is a geometric series with the first term a = 8, and the common ratio r = 1/5. To find the sum of the series, we can use the formula for the sum of an infinite geometric series:

S = a / (1 - r)

So, in this case, the sum of the series is:

S = 8 / (1 - 1/5) = 8 / (4/5) = 8 * (5/4) = 40/4 = 10.

Therefore, the sum of the series is 10.

Answer 2

10

Explanation:

A geometric series is the sum of the terms of a geometric sequence.

`\boxed{\begin{minipage}{5.5 cm}\underline{Sum of an infinite geometric series}\\\\$S_(\infty)=(a)/(1-r)$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $r$ is the common ratio.\\\end{minipage}}`

Given geometric series:

• 8 + 1.6 + 0.32 + 0.064 + ...

The ellipsis means to "continue in this pattern". Therefore, find the sum to infinity of the given geometric series.

From inspection of the sequence, the first term is 8:

`\implies a=8`

To find the common ratio, divide consecutive terms:

`\implies r=(a_2)/(a_1)=(1.6)/(8)=0.2`

Substitute the found values of a and r into the geometric series formula:

`\implies S_(\infty)=(8)/(1-0.2)=(8)/(0.8)=10`

Therefore, the sum of the given geometric series is 10.