Geometric Sequence Formulas Explained

A geometric sequence is a sequence of numbers in which each term is obtained by multiplying the previous term by the same fixed number. This fixed number is called the common ratio. Geometric sequences appear in many areas of mathematics and can also be used to describe real-world situations involving repeated growth or decrease, such as compound interest, population changes, depreciation, and repeated percentage changes.

Understanding geometric sequence formulas makes it easier to find any term in a sequence, calculate the sum of several terms, and analyze how a sequence behaves as the number of terms increases. In this article, we will explore the main geometric sequence formulas, explain what each symbol means, and look at how these formulas are applied.

What Is a Geometric Sequence?

A geometric sequence is an ordered list of numbers in which the ratio between any two consecutive terms remains constant.

For example:

2, 6, 18, 54, 162, …

Each term is obtained by multiplying the previous term by 3:

• 2 × 3 = 6

• 6 × 3 = 18

• 18 × 3 = 54

• 54 × 3 = 162

Therefore, the common ratio is 3.

Another example is:

80, 40, 20, 10, 5, …

Here, every term is multiplied by 1/2 to obtain the next term. Therefore, the common ratio is 1/2.

A geometric sequence can increase, decrease, remain constant, or alternate between positive and negative values depending on its common ratio.

Common Ratio of a Geometric Sequence

The common ratio is the fixed number by which one term is multiplied to obtain the next term.

It is usually represented by r.

The formula for the common ratio is:

r = second term ÷ first term

More generally:

r = aₙ ÷ aₙ₋₁

where aₙ is a term and aₙ₋₁ is the term immediately before it.

For example, consider the sequence:

5, 15, 45, 135, …

The common ratio is:

r = 15 ÷ 5 = 3

Checking the next pair:

45 ÷ 15 = 3

Since the ratio remains constant, the sequence is geometric.

General Form of a Geometric Sequence

A geometric sequence can be written in the general form:

a, ar, ar², ar³, ar⁴, …

Here:

• a = first term

• r = common ratio

• ar = second term

• ar² = third term

• ar³ = fourth term

• ar⁴ = fifth term

For example, if the first term is 4 and the common ratio is 2, the sequence is:

4, 8, 16, 32, 64, …

The terms can be written as:

• First term = 4

• Second term = 4 × 2

• Third term = 4 × 2²

• Fourth term = 4 × 2³

• Fifth term = 4 × 2⁴

This pattern leads directly to the formula for finding the nth term.

Formula for the nth Term of a Geometric Sequence

The most important formula for a geometric sequence is the nth-term formula:

aₙ = arⁿ⁻¹

where:

• aₙ = nth term

• a = first term

• r = common ratio

• n = position of the term

The exponent is n − 1 because the first term has no multiplication by the common ratio.

For the first term:

a₁ = ar⁰ = a

For the second term:

a₂ = ar¹ = ar

For the third term:

a₃ = ar²

This pattern continues for every term.

Example of Finding the nth Term

Consider the sequence:

3, 12, 48, 192, …

The first term is:

a = 3

The common ratio is:

r = 12 ÷ 3 = 4

Suppose we want to find the 7th term.

Using:

aₙ = arⁿ⁻¹

we get:

a₇ = 3 × 4⁶

Since:

4⁶ = 4096

therefore:

a₇ = 3 × 4096 = 12,288

So, the seventh term is 12,288.

How to Find the Common Ratio

When the first few terms of a geometric sequence are given, divide any term by the term immediately before it.

The formula is:

r = aₙ ÷ aₙ₋₁

For example:

7, 21, 63, 189, …

The common ratio is:

r = 21 ÷ 7 = 3

We can confirm it using another pair:

63 ÷ 21 = 3

Therefore, the common ratio is 3.

A sequence is geometric only when the ratio between consecutive terms is constant.

Sum of the First n Terms of a Geometric Sequence

Sometimes we need to add several terms of a geometric sequence. Instead of adding every term individually, we can use the formula for the sum of the first n terms.

When r ≠ 1, the formula is:

Sₙ = a(rⁿ − 1) ÷ (r − 1)

An equivalent form is:

Sₙ = a(1 − rⁿ) ÷ (1 − r)

Both formulas give the same result.

Here:

• Sₙ = sum of the first n terms

• a = first term

• r = common ratio

• n = number of terms

The form you choose often depends on whether the common ratio is greater than or less than 1.

Example of Finding the Sum

Consider:

2, 6, 18, 54, 162

Find the sum of the first five terms.

Here:

a = 2

r = 3

n = 5

Using:

Sₙ = a(rⁿ − 1) ÷ (r − 1)

we get:

S₅ = 2(3⁵ − 1) ÷ (3 − 1)

Since:

3⁵ = 243

we get:

S₅ = 2(243 − 1) ÷ 2

S₅ = 242

Therefore, the sum of the first five terms is 242.

We can verify this by direct addition:

2 + 6 + 18 + 54 + 162 = 242

Sum Formula When the Common Ratio Is 1

The standard geometric sum formula involves division by r − 1 or 1 − r. Therefore, it cannot be used directly when r = 1.

If the common ratio is 1, every term is the same.

For example:

5, 5, 5, 5, 5, …

If there are n terms, the sum is simply:

Sₙ = na

For example, if there are 8 terms and every term is 5:

S₈ = 8 × 5 = 40

Geometric Sequence With a Fractional Common Ratio

A geometric sequence does not have to increase. If the common ratio lies between 0 and 1, the terms generally decrease in magnitude.

For example:

100, 50, 25, 12.5, 6.25, …

Here:

r = 1/2

Each term is half of the previous term.

The nth-term formula is still:

aₙ = arⁿ⁻¹

For example, the fourth term is:

a₄ = 100(1/2)³

a₄ = 100/8

a₄ = 12.5

This type of sequence is useful for describing repeated decreases, such as depreciation or repeated reduction.

Geometric Sequence With a Negative Common Ratio

A geometric sequence can also have a negative common ratio. In this case, the signs of the terms alternate.

For example:

4, −8, 16, −32, 64, …

The common ratio is:

r = −8 ÷ 4 = −2

Each term is obtained by multiplying the previous term by −2.

Using the nth-term formula:

aₙ = arⁿ⁻¹

we can find any term.

For example:

a₅ = 4(−2)⁴

a₅ = 4 × 16

a₅ = 64

The negative ratio causes the sequence to alternate between positive and negative values.

Sum of a Finite Geometric Sequence

A geometric sequence containing a fixed number of terms is called a finite geometric sequence.

For example:

3, 9, 27, 81

has four terms.

Its sum can be calculated using:

Sₙ = a(rⁿ − 1) ÷ (r − 1)

For this sequence:

a = 3

r = 3

n = 4

Therefore:

S₄ = 3(3⁴ − 1) ÷ (3 − 1)

S₄ = 3(81 − 1) ÷ 2

S₄ = 120

So the sum is 120.

Infinite Geometric Sequence

A geometric sequence can continue indefinitely. When we add all of its terms, we have an infinite geometric series.

An infinite geometric series has a finite sum only when:

|r| < 1

The formula is:

S∞ = a ÷ (1 − r)

where:

• S∞ = sum to infinity

• a = first term

• r = common ratio

For example:

10 + 5 + 2.5 + 1.25 + …

Here:

a = 10

r = 1/2

Therefore:

S∞ = 10 ÷ (1 − 1/2)

S∞ = 10 ÷ 1/2

S∞ = 20

Although there are infinitely many terms, their total approaches 20.

When Does an Infinite Geometric Series Have a Sum?

The condition for a finite infinite sum is:

|r| < 1

This means the common ratio must lie between −1 and 1.

For example:

r = 1/2

The terms become smaller and smaller, so the sum approaches a finite value.

However, if:

r = 2

the terms continue growing:

1, 2, 4, 8, 16, …

The sum does not approach a finite number.

If:

r = −2

the terms alternate and increase in magnitude:

1, −2, 4, −8, 16, …

This also does not have a finite sum.

Finding a Missing Term in a Geometric Sequence

The formulas can also help when one term in a geometric sequence is unknown.

For example:

5, 15, ?, 135

The common ratio is:

r = 15 ÷ 5 = 3

Therefore, the missing term is:

15 × 3 = 45

The sequence becomes:

5, 15, 45, 135

Another way to understand this is by using the nth-term formula.

Geometric Mean

The geometric mean connects two positive numbers using a geometric sequence.

If three positive numbers form a geometric sequence:

a, G, b

then:

G² = ab

Therefore:

G = √ab

For example, find the geometric mean between 4 and 16.

Using:

G = √(4 × 16)

G = √64

G = 8

Therefore:

4, 8, 16

is a geometric sequence.

The geometric mean is particularly useful when dealing with multiplicative relationships and proportional changes.

Applications of Geometric Sequences

Geometric sequences are not limited to textbook mathematics. They are useful whenever a quantity changes by the same multiplication factor repeatedly.

Compound Interest

Compound interest can involve geometric growth because money is repeatedly multiplied by a growth factor.

For example, if an amount increases by a fixed percentage each year, the balance can follow a geometric pattern.

Population Growth

If a population grows by a constant percentage over equal periods, its size can be modeled using a geometric sequence.

Depreciation

When the value of an asset decreases by a fixed percentage each year, the remaining value follows a geometric pattern.

Repeated Discounts

Applying the same percentage reduction repeatedly also produces geometric behavior.

Scientific and Physical Models

Geometric sequences can appear in models involving repeated scaling, radioactive processes, wave amplitudes, and other phenomena where a quantity changes by a constant factor.

Important Geometric Sequence Formulas

The main formulas can be summarized as follows.

Common ratio:

r = aₙ ÷ aₙ₋₁

nth term:

aₙ = arⁿ⁻¹

Sum of the first n terms:

Sₙ = a(rⁿ − 1) ÷ (r − 1)

or:

Sₙ = a(1 − rⁿ) ÷ (1 − r)

Sum when r = 1:

Sₙ = na

Sum to infinity when |r| < 1:

S∞ = a ÷ (1 − r)

Geometric mean between two positive numbers:

G = √ab

These formulas provide the basic mathematical tools needed to work with geometric sequences and series.

Conclusion

Geometric sequences are sequences in which each term is obtained by multiplying the previous term by a constant common ratio. Once the first term and common ratio are known, the nth-term formula can be used to find any term in the sequence. The finite sum formula makes it possible to add many terms efficiently, while the infinite sum formula can be used when the absolute value of the common ratio is less than 1.

The key to solving geometric sequence problems is identifying the first term, common ratio, and number of terms before selecting the appropriate formula. With these basic ideas and formulas, geometric sequences become much easier to understand and apply in mathematics and real-world situations.