Question

Find the sum of the first 9 terms in the following geometric series.

Do not round your answer.
64+32+16+...64+32+16+...64, plus, 32, plus, 16, plus, point, point, point

Answer 1

Answer: 127.75.

(Got it wrong, clicked on help, and it showed me the answer. Done.)

Answer 2

The sum of the first 9 terms is
`S_9=(511)/(4)`.

Explanation:

To find the sum of the first
`S_n` terms of a geometric sequence use the formula

`S_n=(a_1(1-r^n))/(1-r)`

where,
`n` is the number of terms,
`a_1` is the first term and
`r` is the common ratio.

To find the common ratio, find the ratio between a term and the term preceding it.

Given the geometric sequence
`64+32+16+...`, the common ratio is

`r=(32)/(64) =(1)/(2)`

and the sum of the first 9 terms is

`S_9=(64(1-((1)/(2))^9))/(1-(1)/(2))\\\\S_9=(64\left(-\left((1)/(2)\right)^9+1\right))/((1)/(2))\\\\S_9=(64\left(-(1)/(2^9)+1\right))/((1)/(2))\\\\S_9=(64\left(-(1)/(512)+1\right))/((1)/(2))\\\\S_9=(64\left(1-(1)/(512)\right)\cdot \:2)/(1)=(128\left(-(1)/(512)+1\right))/(1)\\\\S_9=(128\cdot (511)/(512))/(1)=(511\cdot \:128)/(512)=(65408)/(512)=(511)/(4)`