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Solution - Geometric Sequences

The common ratio is: r=0.4
r=-0.4
The sum of this series is: s=5568
s=5568
The general form of this series is: an=80000.4n1
a_n=8000*-0.4^(n-1)
The nth term of this series is: 8000,3200,1280.0000000000002,512.0000000000001,204.80000000000004,81.92000000000002,32.768000000000015,13.107200000000004,5.242880000000003,2.097152000000001
8000,-3200,1280.0000000000002,-512.0000000000001,204.80000000000004,-81.92000000000002,32.768000000000015,-13.107200000000004,5.242880000000003,-2.097152000000001

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=32008000=0.4

a3a2=12803200=0.4

a4a3=5121280=0.4

The common ratio (r) of the sequence is constant and equals the quotient of two consecutive terms.
r=0.4

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=8,000, the common ratio: r=-0.4, and the number of elements n=4 into the geometric series sum formula:

s4=8000*((1--0.44)/(1--0.4))

s4=8000*((1-0.025600000000000005)/(1--0.4))

s4=8000*(0.9744/(1--0.4))

s4=8000*(0.9744/1.4)

s4=80000.6960000000000001

s4=5568.000000000001

3. Find the general form

an=arn1

To find the general form of the series, plug the first term: a=8,000 and the common ratio: r=0.4 into the formula for geometric series:

an=80000.4n1

4. Find the nth term

Use the general form to find the nth term

a1=8000

a2=a1·rn1=80000.421=80000.41=80000.4=3200

a3=a1·rn1=80000.431=80000.42=80000.16000000000000003=1280.0000000000002

a4=a1·rn1=80000.441=80000.43=80000.06400000000000002=512.0000000000001

a5=a1·rn1=80000.451=80000.44=80000.025600000000000005=204.80000000000004

a6=a1·rn1=80000.461=80000.45=80000.010240000000000003=81.92000000000002

a7=a1·rn1=80000.471=80000.46=80000.0040960000000000015=32.768000000000015

a8=a1·rn1=80000.481=80000.47=80000.0016384000000000006=13.107200000000004

a9=a1·rn1=80000.491=80000.48=80000.0006553600000000003=5.242880000000003

a10=a1·rn1=80000.4101=80000.49=80000.0002621440000000001=2.097152000000001

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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