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Solution - Geometric Sequences

The common ratio is: r=1.8181818181818181
r=1.8181818181818181
The sum of this series is: s=92
s=-92
The general form of this series is: an=331.8181818181818181n1
a_n=-33*1.8181818181818181^(n-1)
The nth term of this series is: 33,60,109.09090909090908,198.3471074380165,360.6311044327573,655.6929171504677,1192.1689402735776,2167.579891406505,3941.054348011827,7165.553360021503
-33,-60,-109.09090909090908,-198.3471074380165,-360.6311044327573,-655.6929171504677,-1192.1689402735776,-2167.579891406505,-3941.054348011827,-7165.553360021503

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=6033=1.8181818181818181

The common ratio (r) of the sequence is constant and equals the quotient of two consecutive terms.
r=1.8181818181818181

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=-33, the common ratio: r=1.8181818181818181, and the number of elements n=2 into the geometric series sum formula:

s2=-33*((1-1.81818181818181812)/(1-1.8181818181818181))

s2=-33*((1-3.305785123966942)/(1-1.8181818181818181))

s2=-33*(-2.305785123966942/(1-1.8181818181818181))

s2=-33*(-2.305785123966942/-0.8181818181818181)

s2=332.818181818181818

s2=92.99999999999999

3. Find the general form

an=arn1

To find the general form of the series, plug the first term: a=33 and the common ratio: r=1.8181818181818181 into the formula for geometric series:

an=331.8181818181818181n1

4. Find the nth term

Use the general form to find the nth term

a1=33

a2=a1·rn1=331.818181818181818121=331.81818181818181811=331.8181818181818181=60

a3=a1·rn1=331.818181818181818131=331.81818181818181812=333.305785123966942=109.09090909090908

a4=a1·rn1=331.818181818181818141=331.81818181818181813=336.010518407212621=198.3471074380165

a5=a1·rn1=331.818181818181818151=331.81818181818181814=3310.92821528584113=360.6311044327573

a6=a1·rn1=331.818181818181818161=331.81818181818181815=3319.86948233789296=655.6929171504677

a7=a1·rn1=331.818181818181818171=331.81818181818181816=3336.12633152344175=1192.1689402735776

a8=a1·rn1=331.818181818181818181=331.81818181818181817=3365.68423913353045=2167.579891406505

a9=a1·rn1=331.818181818181818191=331.81818181818181818=33119.42588933369173=3941.054348011827

a10=a1·rn1=331.8181818181818181101=331.81818181818181819=33217.1379806067122=7165.553360021503

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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