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Solution - Geometric Sequences

The common ratio is: r=1.0444444444444445
r=-1.0444444444444445
The sum of this series is: s=4
s=-4
The general form of this series is: an=901.0444444444444445n1
a_n=90*-1.0444444444444445^(n-1)
The nth term of this series is: 90,94,98.17777777777779,102.54123456790124,107.09862277091909,111.85856156073773,116.83005318565941,122.02249999391094,127.44572221586256,133.10997653656756
90,-94,98.17777777777779,-102.54123456790124,107.09862277091909,-111.85856156073773,116.83005318565941,-122.02249999391094,127.44572221586256,-133.10997653656756

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=9490=1.0444444444444445

The common ratio (r) of the sequence is constant and equals the quotient of two consecutive terms.
r=1.0444444444444445

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=90, the common ratio: r=-1.0444444444444445, and the number of elements n=2 into the geometric series sum formula:

s2=90*((1--1.04444444444444452)/(1--1.0444444444444445))

s2=90*((1-1.0908641975308644)/(1--1.0444444444444445))

s2=90*(-0.09086419753086439/(1--1.0444444444444445))

s2=90*(-0.09086419753086439/2.0444444444444443)

s2=900.04444444444444454

s2=4.000000000000009

3. Find the general form

an=arn1

To find the general form of the series, plug the first term: a=90 and the common ratio: r=1.0444444444444445 into the formula for geometric series:

an=901.0444444444444445n1

4. Find the nth term

Use the general form to find the nth term

a1=90

a2=a1·rn1=901.044444444444444521=901.04444444444444451=901.0444444444444445=94

a3=a1·rn1=901.044444444444444531=901.04444444444444452=901.0908641975308644=98.17777777777779

a4=a1·rn1=901.044444444444444541=901.04444444444444453=901.1393470507544583=102.54123456790124

a5=a1·rn1=901.044444444444444551=901.04444444444444454=901.1899846974546566=107.09862277091909

a6=a1·rn1=901.044444444444444561=901.04444444444444455=901.2428729062304191=111.85856156073773

a7=a1·rn1=901.044444444444444571=901.04444444444444456=901.2981117020628823=116.83005318565941

a8=a1·rn1=901.044444444444444581=901.04444444444444457=901.3558055554878994=122.02249999391094

a9=a1·rn1=901.044444444444444591=901.04444444444444458=901.4160635801762507=127.44572221586256

a10=a1·rn1=901.0444444444444445101=901.04444444444444459=901.4789997392951952=133.10997653656756

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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