Geometric sequences are a fundamental concept in mathematics that appear in various fields, from finance to physics. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This sequence type can model exponential growth or decay, making it incredibly useful for understanding real-world phenomena like population growth, radioactive decay, and interest calculations.

Let’s look into the geometric sequence formulas step by step and look at some solved examples to help clarify the concepts.

What Are Geometric Sequence Formulas?

Geometric sequence formulas are mathematical expressions used to find specific terms in a geometric sequence and to calculate the sum of terms within such a sequence. These formulas help us understand the pattern and behaviour of numbers in a sequence where each term is obtained by multiplying the previous term by a constant called the common ratio.

Geometric Sequence Formulas

Let us look at the Key Formulas of Geometric Sequence essential for solving various mathematical and real-world problems:

1. Formula for the nth Term of a Geometric Sequence

We consider the sequence to be a, ar, ar², ar³,…. Its first term is a (or ar^1-1 ), its second term is ar (or ar^2-1 ), and its third term is ar² (or ar^3-1 ). The formula to find the n^th term ( a_n ​) of a geometric sequence is:

a_n = a · r^{n – 1}

Where,

• a_n ​ = nth term of the sequence
• a₁ ​ = first term of the sequence
• r = common ratio (the factor by which we multiply to get the next term)
• n = position of the term in the sequence (term number)

2. Sum of the First n Terms of a Geometric Sequence (Finite Geometric Series)

The to find the sum ( S_n ​) of the first ‘n’ terms of the geometric sequence a, ar, ar² , ar³ , . . . is:

[Tex]S_n = a_1 \frac{r^n – 1}{r – 1}[/Tex]

3. Sum of an Infinite Geometric Series

For an infinite geometric series where the absolute value of the common ratio is less than 1 (∣r∣ < 1) i.e. the Convergence Criteria, the sum is:

S_∞ = a₁/(1 – r)

Where:

• S = the sum of the series
• a = the first term
• r = the common ratio

Note: This formula is valid only when ∣r∣ < 1. If r > 1, the infinite geometric sequence diverges, meaning its sum cannot be determined.

4. Common Ratio (Given Two Terms)

If you know two consecutive terms a_n and a_n+1 of a geometric sequence, the common ratio (r) can be found using:

[Tex]r = \frac{a_{n+1}}{a_n}[/Tex]

5. Geometric Mean

The geometric mean of two numbers a and b is:

Geometric Mean = ​[Tex]\sqrt{a \cdot b}[/Tex]

This value is particularly useful in various applications such as growth rates and finance.

6. Product of Terms in a Geometric Sequence

For a geometric sequence with n terms a₁, a₂, a₃, . . . ,a_n ​with common ratio r, the product of all the terms is given by:

P = (a₁ ​⋅ a_n​)^n/2

Examples of Geometric Sequence Formulas

Let us look at some of the examples to better understand these Forumulas.

Example 1: Find the 5^th term of a geometric sequence where the first term a₁ is 3 and the common ratio r is 2.

Solution:

The formula for the n^th term of a geometric sequence is:

a_n = a₁ · r^n-1

Here, a₁ = 3, r = 2, and n = 5.

a₅ = 3 · 2^5-1

a₅ = 3 · 2⁴

a₅ = 3 · 16

a₅ = 48

So, the 5^th term is 48.

Example 2: Find the sum of the first 4 terms of a geometric sequence where the first term a_1 is 2 and the common ratio r is 3.

Solution:

The formula for the sum of the first n terms S_n of a geometric sequence is:

[Tex]S_n = a_1 \frac{r^n – 1}{r – 1}[/Tex]

Here, a₁ = 2, r = 3, and n = 4.

[Tex]S_4 = 2 \frac{3^4 – 1}{3 – 1}[/Tex]

⇒ [Tex]S_4 = 2 \cdot \frac{81 – 1}{2}[/Tex]

⇒ S₄ = 2 · 80/2 = 2 · 40 = 80

So, the sum of the first 4 terms is 80.

Example 3: Find the common ratio of a geometric sequence where the 2^nd term is 12 and the 5^th term is 324.

Solution:

The formula for the n^th term of a geometric sequence is:

[Tex]a_n = a_1 · r^{(n-1)}[/Tex]

Let a₂ = 12 and a₅ = 324.

⇒ a₂ = a₁ · r¹

⇒ 12 = a₁ · r

⇒ a₁ = 12/r

For the 5th term:

a₅ = a₁ · r⁴

[Tex]\Rightarrow 324 = \left(\frac{12}{r}\right) · r^4[/Tex]

⇒ 324 = 12 · r³

⇒ r³ = 324/12

⇒ r³ = 27

⇒ r = ∛27 = 3

So, the common ratio is 3.

Example 4: Find the sum to infinity of a geometric series where the first term a₁ is 5 and the common ratio r is 1/3.

Solution:

The formula for the sum to infinity S_∞ of a geometric series is:

S_∞ = a₁/(1 – r)

Here, a₁ = 5 and r = 1/3.

S_∞ = 5/[1 – (1/3)]

⇒ S_∞ = 5/[2/3]

⇒ S_∞ = 5 · (3/2)

⇒ S_∞ = 15/2 = 7.5

So, the sum to infinity is 7.5.

behaviour

Applications of Geometric Sequences

Geometric sequences are not just theoretical concepts but have numerous practical applications across different fields. Here are some of the key areas where geometric sequences are applied:

• Compound Interest: The formula for compound interest, which shows how an investment grows over time, is derived from geometric sequences.
• Annuities: Regular payments or receipts over time, like in annuities, also use geometric sequences to calculate their present and future values.
• Radioactive Decay: The decay of radioactive substances follows a geometric sequence. The amount of substance remaining after each half-life period decreases geometrically.
• Signal Processing: In engineering, particularly in signal processing, geometric sequences are used to model and analyze signals and systems.
• Drug Dosage: The dosage of certain medications administered in regular intervals can follow a geometric sequence, particularly when dealing with drugs that have a constant rate of absorption or elimination.

Geometric Sequence Formulas – FAQs

What is a geometric sequence?

A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio.

How do you find the n^th term of a geometric sequence?

The nth term of a geometric sequence can be found using the formula: a_n = a · r^{n – 1} where a is the first term, r is the common ratio, and n is the term number.

What is the common ratio in a geometric sequence?

The common ratio in a geometric sequence is the fixed number that each term is multiplied by to get the next term. It can be calculated by dividing any term by the previous term.

What is the difference between an arithmetic sequence and a geometric sequence?

In an arithmetic sequence, each term is obtained by adding a fixed number called the common difference to the previous term. In a geometric sequence, each term is obtained by multiplying the previous term by a fixed number called the common ratio.

What are some common applications of geometric sequences?

Geometric sequences are used in various fields such as finance (to calculate compound interest), computer science (in algorithms and data structures), and physics (in wave patterns and decay processes).