# Applications of Geometric Sequences and Series

**This page originally created by Jatin P and Vishakan.S (2021)**

**Geometric Sequence:**

A geometric sequence is an organized list of numbers where each term is found being multiplied by the previous one by a fixed non-zero number. This is better know as a common ratio r.

**Geometric Series:**

A geometric series is the sum of the terms of a geometric sequence. If the number of terms in the geometric sequence is finite, the sum of terms is called a finite geometric series. If not, the series is infinite.

**Applications:**

An application of geometric sequences and series are real life scenarios where you will see those patterns occur. Geometric sequences and series play important roles in physics, engineering, biology, economics, computer science, and finance.

## Applications of Geometric Sequences[edit]

### Examples[edit]

#### Geometric Ratios[edit]

The general form of **Geometric Progression** is:

#### nTH Term Formula[edit]

#### Summary[edit]

A geometric sequence is a sequence where each term is calculated by multiplying the previous term by a fixed number.

- a first term, a,
- a common ratio, r.

When the terms of the sequence are added together, this creates a geometric series. There are formulas for calculating the amount terms in a geometric sequence,an infinite geometric series can tend towards a finite number.

## Applications of Geometric Series[edit]

### Examples[edit]

#### Repeating Decimals[edit]

A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example: 0.7777⋯= 7/10 + 7/100 +7/1000 + 7/10000 + ...

The formula for the sum of a geometric series can be used to convert the decimal to a fraction: 0.7777⋯= a/(1-r) = (7/10)/(1-1/10) =(7/10)/(9/10) =(7/10)(10/9) =7/9

Another example is: 0.9999...= a/(1-r) =(9/10)/(1-1/10) =(9/10)/(9/10) =(9/10)(10/9) =9/9 =1

#### Finance[edit]

Geometric series determine how interest-bearing investments accrue with time.You put away a monthly amount m which accrues interest at an annual rate R and a monthly rate r=R/12. After a month you have earned rm interest, and deposited another increment of m:

After another month, you earn interest of r[m(1+r)+m] on your balance, and make another deposit m:

After n months:

We can write this sum in closed form using the formula G(a,n):

This equation shows that your investment is proportional to the monthly payment. While the investment grows linearly with the monthly payment, it grows exponentially with the rate r and with the time n. The term depending on the rate can be thought of a monetary magnification factor M which depends on r and n, or more conveniently, on R=12r and n=12y:

#### Archimedes' Quadrature of the Parabola[edit]

Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. His method was to dissect the area into an infinite number of triangles.

Archimedes' Theorem says that the total area under the parabola is 4/3 the area of the biggest(blue) triangle. He determined that each green triangle has the area of 1/8 of the blue triangle, while the yellow triangle has 1/8 of the green triangle, and so forth. Assuming the biggest triangle has an area of 1:

The first term represents the area of the biggest triangle, the second term is for for the both second biggest triangles(the one in green), and so on. Simplifying the fractions gives:

This is a geometric series with a common ratio of 1/4.

## Resources[edit]

https://www.youtube.com/watch?v=38M0-zqbrJ8&ab_channel=StudyForce

https://en.wikipedia.org/wiki/Geometric_series#Applications

https://en.wikipedia.org/wiki/Arithmetico%E2%80%93geometric_sequence

https://mathmaine.com/2014/05/08/summary-geometric-sequences-and-series/