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Solution - Geometric Sequences

The common ratio is: r=−0.1111111111111111
r=-0.1111111111111111
The sum of this series is: s=−656
s=-656
The general form of this series is: an=−729⋅−0.1111111111111111n−1
a_n=-729*-0.1111111111111111^(n-1)
The nth term of this series is: −729,81,−9,0.9999999999999998,−0.11111111111111109,0.012345679012345677,−0.0013717421124828527,0.00015241579027587253,−1.693508780843028E−05,1.8816764231589197E−06
-729,81,-9,0.9999999999999998,-0.11111111111111109,0.012345679012345677,-0.0013717421124828527,0.00015241579027587253,-1.693508780843028E-05,1.8816764231589197E-06
See steps

Other Ways to Solve

Geometric Sequences

Step-by-step explanation

1. Find the common ratio

Find the common ratio by dividing any term in the sequence by the term that comes before it:

a2a1=81−729=−0.1111111111111111

a3a2=−981=−0.1111111111111111

a4a3=1−9=−0.1111111111111111

The common ratio (r) of the sequence is constant and equals the quotient of two consecutive terms.
r=−0.1111111111111111

2. Find the sum

5 additional steps

sn=a*((1-rn)/(1-r))

To find the sum of the series, plug the first term: a=-729, the common ratio: r=-0.1111111111111111, and the number of elements n=4 into the geometric series sum formula:

s4=-729*((1--0.11111111111111114)/(1--0.1111111111111111))

s4=-729*((1-0.00015241579027587256)/(1--0.1111111111111111))

s4=-729*(0.9998475842097241/(1--0.1111111111111111))

s4=-729*(0.9998475842097241/1.1111111111111112)

s4=−729⋅0.8998628257887517

s4=−656

3. Find the general form

an=a⋅rn−1

To find the general form of the series, plug the first term: a=−729 and the common ratio: r=−0.1111111111111111 into the formula for geometric series:

an=−729⋅−0.1111111111111111n−1

4. Find the nth term

Use the general form to find the nth term

a1=−729

a2=a1·rn−1=−729⋅−0.11111111111111112−1=−729⋅−0.11111111111111111=−729⋅−0.1111111111111111=81

a3=a1·rn−1=−729⋅−0.11111111111111113−1=−729⋅−0.11111111111111112=−729⋅0.012345679012345678=−9

a4=a1·rn−1=−729⋅−0.11111111111111114−1=−729⋅−0.11111111111111113=−729⋅−0.001371742112482853=0.9999999999999998

a5=a1·rn−1=−729⋅−0.11111111111111115−1=−729⋅−0.11111111111111114=−729⋅0.00015241579027587256=−0.11111111111111109

a6=a1·rn−1=−729⋅−0.11111111111111116−1=−729⋅−0.11111111111111115=−729⋅−1.6935087808430282E−05=0.012345679012345677

a7=a1·rn−1=−729⋅−0.11111111111111117−1=−729⋅−0.11111111111111116=−729⋅1.8816764231589202E−06=−0.0013717421124828527

a8=a1·rn−1=−729⋅−0.11111111111111118−1=−729⋅−0.11111111111111117=−729⋅−2.090751581287689E−07=0.00015241579027587253

a9=a1·rn−1=−729⋅−0.11111111111111119−1=−729⋅−0.11111111111111118=−729⋅2.3230573125418763E−08=−1.693508780843028E−05

a10=a1·rn−1=−729⋅−0.111111111111111110−1=−729⋅−0.11111111111111119=−729⋅−2.581174791713196E−09=1.8816764231589197E−06

Why learn this

Geometric sequences are commonly used to explain concepts in mathematics, physics, engineering, biology, economics, computer science, finance, and more, making them a very useful tool to have in our toolkits. One of the most common applications of geometric sequences, for example, is calculating earned or unpaid compound interest, an activity most commonly associated with finance that could mean earning or losing a lot of money! Other applications include, but are certainly not limited to, calculating probability, measuring radioactivity over time, and designing buildings.

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