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Solution - Geometric Sequences

The common ratio is: r=4
r=-4
The sum of this series is: s=1428
s=1428
The general form of this series is: an=284n1
a_n=-28*-4^(n-1)
The nth term of this series is: 28,112,448,1792,7168,28672,114688,458752,1835008,7340032
-28,112,-448,1792,-7168,28672,-114688,458752,-1835008,7340032

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=11228=4

a3a2=448112=4

a4a3=1792448=4

The common ratio (r) of the sequence is constant and equals the quotient of two consecutive terms.
r=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=-28, the common ratio: r=-4, and the number of elements n=4 into the geometric series sum formula:

s4=-28*((1--44)/(1--4))

s4=-28*((1-256)/(1--4))

s4=-28*(-255/(1--4))

s4=-28*(-255/5)

s4=2851

s4=1428

3. Find the general form

an=arn1

To find the general form of the series, plug the first term: a=28 and the common ratio: r=4 into the formula for geometric series:

an=284n1

4. Find the nth term

Use the general form to find the nth term

a1=28

a2=a1·rn1=28421=2841=284=112

a3=a1·rn1=28431=2842=2816=448

a4=a1·rn1=28441=2843=2864=1792

a5=a1·rn1=28451=2844=28256=7168

a6=a1·rn1=28461=2845=281024=28672

a7=a1·rn1=28471=2846=284096=114688

a8=a1·rn1=28481=2847=2816384=458752

a9=a1·rn1=28491=2848=2865536=1835008

a10=a1·rn1=284101=2849=28262144=7340032

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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