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## Geometric Series

A **geometric series** is a series whose related sequence is geometric. It results from adding the terms of a geometric sequence .

## Finite Geometric Series

To find the sum of a finite geometric series, use the formula,

## Infinite Geometric Series

## Geometric Series Formula

Before knowing the geometric series formula, let us recall what is a geometric series. It is a series where the ratios of every two consecutive terms are the same. The geometric series formula includes:

- Formula to find the n
^{th }term of a geometric sequence. - Finding the sum of a finite geometric series.
- Finding the sum of an infinite geometric series.

## What Is a Geometric Series Formula?

The geometric series formula refers to the formula that gives the sum of a finite geometric sequence, the sum of an infinite geometric series, and the n^{th} term of a geometric sequence. The sequence is of the form {a, ar, ar^{2}, ar^{3}, …….} where, a is the first term, and r is the “common ratio”.

### Geometric Series Formula

The formulas for a geometric series include the formulas to find the n^{th} term, the sum of n terms, and the sum of infinite terms. Let us consider a geometric series whose first term is a and common ratio is r.

a + ar + ar^{2} + ar^{3} + …

Formula 1: The n^{th} term of a geometric sequence is,

n^{th} term = a r^{n-1}

Where,

- a is the first term
- r is the common ratio for all the terms
- n is the number of terms.

Formula 2: The sum formula of a finite geometric series a + ar + ar^{2} + ar^{3} + … + a r^{n-1} is

Sum of n terms = a (1 – r^{n}) / (1 – r)

where,

- a is the first term
- r is the common ratio for all the terms
- n is the number of terms.

Formula 3: The sum formula of an infinite geometric series a + ar + ar^{2} + ar^{3} + … is

Sum of infinite geometric series = a / (1 – r)

where,

- a is the first term
- r is the common ratio for all the terms

## Applications of Geometric Series Formula

Geometric series formulas are used throughout mathematics. These have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.

Let us learn the applications of the formula for a geometric series in the upcoming sections.

## Examples Using Formula for a Geometric Series

**Example 1: **Find the 10^{th} term of the geometric sequence 1, 4, 16, 64, …

**Solution:**

To find: The 10^{th} term of the given geometric sequence.

In the given sequence,

The first term, a = 1.

The common ratio, r = 4 / 1 (or) 16 / 4 (or) 64 / 16 = 4.

Using the formulas of a geometric series, the n^{th} term is found using:

n^{th} term = a r^{n-1}

Substitute n = 10, a = 1, and r = 4 in the above formula:

10^{th} term = 1 × 4^{10-1} = 4^{9} = 262,144

**Answer: The 10 ^{th} term of the given geometric sequence = 262,144.**

**Example 2: **Find the sum of the following geometric series: i) 1 + (1/3) + (1/9) + … + (1/2187) ii) 1 + (1/3) + (1/9) + … using the formula for a geometric series.

**Solution:**

To find: The sum of the given two geometric series.

In both of the given series,

the first term, a = 1.

The common ratio, r = 1 / 3.

i) In the given series,

n^{th} term = 1 / 2187

a r^{n-1} = 1 / 3^{7}

1 × ( 1 / 3) ^{n-1} = 3^{-7}

3^{-n + 1 } = 3^{-7}

-n + 1 = -7

n = 8

So we need to find the sum of the first 8 terms of the given series.

Using the sum of the finite geometric series formula:

Sum of n terms = a (1 – r^{n}) / (1 – r)

Sum of 8 terms = 1 ( 1 – (1/3)^{8} ) / (1 – 1/3)

= (1 – (1 / 6561)) / (2 / 3)

= (6560 / 6561) × (3 / 2)

= 3280 / 2187

ii) The given series is an infinite geometric series.

Using the sum of the infinite geometric series formula:

Sum of infinite geometric series = a / (1 – r)

Sum of the given infinite geometric series

= 1 / (1 – (1/3))

= 1 / (2 / 3)

= 3 / 2

**Answer: i) Sum = 3280 / 2187 and ii) Sum = 3 / 2**

**Example 3:** Calculate the sum of geometric series if a = 5, r = 1.5 and n = 10.

**Solution:**

To find: the sum of geometric series

Given: a = 5, r = 1.5, n = 10

s_{n} = a(1−r^{n})/(1−r)

The sum of ten terms is given by S_{10 }= 5(1−(1.5)^{10})/(1−1.5)

= 566.65

**Answer: The sum of the geometric series is 566.65.**

## FAQs on Geometric Series Formula

### What Are the Geometric Series Formulas in Math?

The geometric series formulas are the formulas that help to calculate the sum of a finite geometric sequence, the sum of an infinite geometric series, and the n^{th} term of a geometric sequence. These formulas are geometric series with first term ‘a’ and common ratio ‘r’ given as,

- n
^{th}term = a r^{n-1} - Sum of n terms = a (1 – r
^{n}) / (1 – r) - Sum of infinite geometric series = a / (1 – r)

### What Is the Infinite Geometric Series Formula?

The sum formula of an infinite geometric series a + ar + ar^{2} + ar^{3} + … can be calculated using the formula, Sum of infinite geometric series = a / (1 – r), where a is the first term, r is the common ratio for all the terms and n is the number of terms.

### How To Use the Geometric Series Formula?

To use the geometric series formula

- Step 1: Check for the given values, a, r and n.
- Step 2: Put the values in the geometric series formula as per the requirement – the sum of a finite geometric sequence, the sum of an infinite geometric series, or the n
^{th}term of a geometric sequence

### What Is ‘r’ in the Geometric Series Formula?

In the geometric series formula, ‘r’ refers to the common ratio. The formulas for geometric series with ‘n’ terms and first term ‘a’ are given as,

- Formula for nth term: n
^{th}term = a r^{n-1} - Sum of n terms = a (1 – r
^{n}) / (1 – r) - Sum of infinite geometric series = a / (1 – r)

## Geometric Sequences and Sums

## Sequence

A Sequence is a set of things (usually numbers) that are in order.

## The Rule

We can also calculate **any term** using the Rule:

x_{n} = ar^{(n-1)}

(We use “n-1” because ar^{0} is for the 1st term)

A Geometric Sequence can also have **smaller and smaller** values:

etc (yes we can have 4 and more dimensions in mathematics). |

Geometric Sequences are sometimes called Geometric Progressions (G.P.’s)

## Summing a Geometric Series

**To sum these:**

a + ar + ar^{2} + … + ar^{(n-1)}

(Each term is ar^{k}, where k starts at 0 and goes up to n-1)

**We can use this handy formula:**

The formula is easy to use … just “plug in” the values of **a**, **r** and **n**

You can check it yourself:

10 + 30 + 90 + 270 = 400

And, yes, it is easier to just add them *in this example*, as there are only 4 terms. But imagine adding 50 terms … then the formula is much easier.

## Using the Formula

Let’s see the formula in action:

Example: Grains of Rice on a Chess Board

And another example, this time with **r** less than 1:

Example: Add up the first 10 terms of the Geometric Sequence that halves each time:

{ 1/2, 1/4, 1/8, 1/16, … }

The values of **a**, **r** and **n** are:

**a = ½**(the first term)**r = ½**(halves each time)**n = 10**(10 terms to add)

So:

## Why Does the Formula Work?

Let’s see **why** the formula works, because we get to use an interesting “trick” which is worth knowing.

**r** must be between (but not including) **−1 and 1**

and **r should not be 0** because the sequence {a,0,0,…} is not geometric

So our infnite geometric series has a **finite sum** when the ratio is less than 1 (and greater than −1)

Let’s bring back our previous example, and see what happens:

Example: Add up ALL the terms of the Geometric Sequence that halves each time:

So there we have it … Geometric Sequences (and their sums) can do all sorts of amazing and powerful things.

## Infinite Geometric Series

Geometric series are one of the simplest examples of infinite series with finite sums.

### Learning Objectives

Calculate the sum of an infinite geometric series and recognize when a geometric series will converge

### Key Takeaways

#### Key Points

- The sum of a geometric series is finite as long as the terms approach zero; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series being infinite.

For an infinite geometric series that converges, its sum can be calculated with the formula s=^{a}/_{1−r}

- .

#### Key Terms

**converge**: Approach a finite sum.**geometric series**: An infinite sequence of summed numbers, whose terms change progressively with a common ratio.

A geometric series is an infinite series whose terms are in a geometric progression, or whose successive terms have a common ratio. If the terms of a geometric series approach zero, the sum of its terms will be finite. As the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series being infinite.

A geometric series with a finite sum is said to converge. A series converges if and only if the absolute value of the common ratio is less than one:

|r|<1

What follows in an example of an infinite series with a finite sum. We will calculate the sum s of the following series:

A similar technique can be used to evaluate any self-similar expression.

A formula can be derived to calculate the sum of the terms of a convergent series. The formula used to sum the first n terms of any geometric series where r≠1 is:

If a series converges, we want to find the sum of not only a finite number of terms, but all of them. If |r|<1, then we see that as n becomes very large, rn becomes very small. We express this by writing that as n→∞ (as n approaches infinity), r^{n}→0

.

Applying r^{n}→0, we can find a new formula for the sum of an infinitely long geometric series:

### Example

Find the sum of the infinite geometric series 64+32+16+8+⋯

First, find r, or the constant ratio between each term and the one that precedes it:

## Applications of Geometric Series

Geometric series have applications in math and science and are one of the simplest examples of infinite series with finite sums.

### Learning Objectives

Apply geometric sequences and series to different physical and mathematical topics

### Key Takeaways

#### Key Points

- A repeating decimal can be viewed as a geometric series whose common ratio is a power of
^{1}/_{10}

- .
- Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line.
- The interior of the Koch snowflake is a union of infinitely many triangles. In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar figure.
- Knowledge of infinite series allows us to solve ancient problems, such as Zeno’s paradoxes.

#### Key Terms

**geometric series**: An infinite sequence of summed numbers, whose terms change progressively with a common ratio.**fractal**: A natural phenomenon or mathematical set that exhibits a repeating pattern that can be seen at every scale.

Geometric series played an important role in the early development of calculus, and continue as a central part of the study of the convergence of series. Geometric series are used throughout mathematics. They have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.

Geometric series are one of the simplest examples of infinite series with finite sums, although not all of them have this property.

### Repeating Decimal

A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example:

That is, a repeating decimal with a repeating part of length n is equal to the quotient of the repeating part (as an integer) and 10^{n}−1.

### Archimedes’ Quadrature of the Parabola

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 states that the total area under the parabola is ^{4}/_{3} of the area of the blue triangle. He determined that each green triangle has ^{1}/_{8} the area of the blue triangle, each yellow triangle has ^{1}/_{8} the area of a green triangle, and so forth. Assuming that the blue triangle has area 1, the total area is an infinite series:

The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives:

This is a geometric series with common ratio ^{1}/_{4}, and the fractional part is equal to ^{1}/_{3}.

### Fractal Geometry

The Koch snowflake is a fractal shape with an interior comprised of an infinite amount of triangles. In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar figure. In the case of the Koch snowflake, its area can be described with a geometric series.

**Constructing the Koch snowflake:** the first four iterations: Each iteration adds a set of triangles to the outside of the shape.

The area inside the Koch snowflake can be described as the union of an infinite number of equilateral triangles. In the diagram above, the triangles added in the second iteration are exactly ^{1}/_{3} the size of a side of the largest triangle, and therefore they have exactly ^{1}/_{9} the area. Similarly, each triangle added in the second iteration has ^{1}/_{9} the area of the triangles added in the previous iteration, and so forth. Taking the first triangle as a unit of area, the total area of the snowflake is:

Thus the Koch snowflake has ^{8}/_{5} of the area of the base triangle.

### Zeno’s Paradoxes

Zeno’s Paradoxes are a set of philosophical problems devised by an ancient Greek philosopher to support the doctrine that the truth is contrary to one’s senses. Simply stated, one of Zeno’s paradoxes says: There is a point, A, that wants to move to another point, B. If A only moves half of the distance between it and point B at a time, it will never get there, because you can continue to divide the remaining space in half forever. Zeno’s mistake is in the assumption that the sum of an infinite number of finite steps cannot be finite. We now know that his paradox is not true, as evidenced by the convergence of the geometric series with r=^{1}/_{2}. This problem has been solved by modern mathematics, which can apply the concept of infinite series to find a sum of the distances traveled.

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