Question

The following are infinite geometric series. For which infinite series could find the sum using a formula?

A) -2+6+(-18)+54+(-162)+...

B) 3+12+48+192+768+...

C) 1+2+4+8+16+...

D) 8+4+2+1+1/2+...

Answer 1

D) 8 + 4 + 2 + 1 + 1/2 + ...

Explanation:

The sum of an infinite geometric series can be found when the absolute value of the common ratio, r, is less than 1.

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

To determine which of the given infinite geometric series can be summed using the formula, calculate the value of r for each series. To do this, divide a term by the previous term.

Given infinite geometric series:

• -2 + 6 + (-18) + 54 + (-162) + ...

`\implies r=(6)/(-2)=-3`

As r is not in the interval −1 < r < 1, the sum of the infinite series cannot be found using the formula.

Given infinite geometric series:

• 3 + 12 + 48 + 192 + 768 + ...

`\implies r=(12)/(3)=4`

As r is not in the interval −1 < r < 1, the sum of the infinite series cannot be found using the formula.

Given infinite geometric series:

• 1 + 2 + 4 + 8 + 16 + ...

`\implies r=(2)/(1)=2`

As r is not in the interval −1 < r < 1, the sum of the infinite series cannot be found using the formula.

Given infinite geometric series:

• 8 + 4 + 2 + 1 + 1/2 + ...

`\implies r=(4)/(8)=(1)/(2)`

As |r| < 1 the sum of the infinite series can be found using the formula.

Therefore, the infinite series for which the sum can be found by using the formula is:

• D) 8 + 4 + 2 + 1 + 1/2 + ...