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Solution - Geometric Sequences

The common ratio is: r=1.0689655172413792
r=-1.0689655172413792
The sum of this series is: s=1
s=-1
The general form of this series is: an=291.0689655172413792n1
a_n=29*-1.0689655172413792^(n-1)
The nth term of this series is: 29,30.999999999999996,33.137931034482754,35.42330558858501,37.86629218090121,40.47776060717026,43.26933030421648,46.25342204933486,49.443313225151044,52.85319689585112
29,-30.999999999999996,33.137931034482754,-35.42330558858501,37.86629218090121,-40.47776060717026,43.26933030421648,-46.25342204933486,49.443313225151044,-52.85319689585112

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=3129=1.0689655172413792

The common ratio (r) of the sequence is constant and equals the quotient of two consecutive terms.
r=1.0689655172413792

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=29, the common ratio: r=-1.0689655172413792, and the number of elements n=2 into the geometric series sum formula:

s2=29*((1--1.06896551724137922)/(1--1.0689655172413792))

s2=29*((1-1.1426872770511294)/(1--1.0689655172413792))

s2=29*(-0.14268727705112938/(1--1.0689655172413792))

s2=29*(-0.14268727705112938/2.068965517241379)

s2=290.06896551724137921

s2=1.9999999999999971

3. Find the general form

an=arn1

To find the general form of the series, plug the first term: a=29 and the common ratio: r=1.0689655172413792 into the formula for geometric series:

an=291.0689655172413792n1

4. Find the nth term

Use the general form to find the nth term

a1=29

a2=a1·rn1=291.068965517241379221=291.06896551724137921=291.0689655172413792=30.999999999999996

a3=a1·rn1=291.068965517241379231=291.06896551724137922=291.1426872770511294=33.137931034482754

a4=a1·rn1=291.068965517241379241=291.06896551724137923=291.2214932961581038=35.42330558858501

a5=a1·rn1=291.068965517241379251=291.06896551724137924=291.3057342131345246=37.86629218090121

a6=a1·rn1=291.068965517241379261=291.06896551724137925=291.3957848485231124=40.47776060717026

a7=a1·rn1=291.068965517241379271=291.06896551724137926=291.492045872559189=43.26933030421648

a8=a1·rn1=291.068965517241379281=291.06896551724137927=291.5949455879080985=46.25342204933486

a9=a1·rn1=291.068965517241379291=291.06896551724137928=291.704941835350036=49.443313225151044

a10=a1·rn1=291.0689655172413792101=291.06896551724137929=291.822524030891418=52.85319689585112

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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