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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

User Salah Atwa
by
7.4k points

2 Answers

1 vote

Answer: 127.75.

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

User Informat
by
7.2k points
4 votes

Answer:

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)

User Matteogll
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7.0k points