Explicit Formulas
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Last updated on October 19, 2025

In mathematics, to find any term of a sequence, we use explicit formulas. In this article, we will be discussing the explicit formulas in detail along with real life applications and common mistakes made by students.

What are Explicit Formulas?

In a sequence to find any term without knowing the previous term, we use the explicit formula. It is a formula used to find the nth term of a sequence based on its position. Let’s learn the explicit formula for different types of sequences:

Type of Sequence Explicit Formula Example Arithmetic Sequence

an = a + (n - 1)d, where a is the first term and d is the common difference

For the sequence: 2, 4, 6, 8,… 
The fifth term, a5 = a + (n - 1)d
Here, n = 5
a = 2
d = 2
a5 = 2 + (5 - 1)  2
= 2 + 4 × 2 = 2 + 8
= 10

Geometric Sequence an = arn-1, where a is the first term and ‘r’ is the common ratio

For the sequence: 2, 6, 18,… 
The fifth term, a5 = arn - 1
Here, n = 5
a = 2
r = 3
a5 = 2 × 3(5 - 1)
= 2 × 34 = 2 × 81 = 162

Harmonic Sequence an = 1/(a + (n - 1)d), 
where a is the first term and d is the common difference.

For the sequence: 1/3, 1/7, 1/11,…

How To Find Explicit Formulas?

To find the nth term of a sequence, we use an explicit formula. The sequence can be arithmetic, geometric, or harmonic.

nth term of an arithmetic sequence -  an = a + (n - 1)d, where
a is the first term

n is the position of the term in the sequence, 

and d is the common difference. Steps used for finding explicit formulas are -

Step 1: First find the first term and the common difference of the sequence

Step 2: Substitute the value of a, n, and d in the explicit formula, an = a + (n - 1)d

Step 3: Simplify the formula to find the nth term

To find the 7th term of the sequence 7, 14, 21, 28, … 

In the given sequence, a = 7 and d = 7(14 - 7 = 7)

The nth term of an arithmetic sequence is: an = a + (n - 1)d

So, the 7th term is: a7 = 7 + (7 - 1)7

= 7+ (6 × 7)

= 7 + 42 = 49

Therefore, the 7th term of the sequence is 49.

Explicit Formula For Arithmetic Sequence

In an arithmetic sequence, the difference between consecutive terms is constant and is called the common difference (d). To find the nth term of an arithmetic sequence, we use the explicit formula: an = a + (n - 1)d. 
Where, 

  • an is the nth term
  • a is the first term
  • d is the common difference

For example, for the arithmetic sequence 2, 5, 8, 11, 14, …, finding the explicit formula

  Here, a = 2

d = 3

The explicit formula of an arithmetic sequence is: an = a + (n - 1)d
Substituting the value of a and d:

an = 2 + (n - 1)3

= 2 + 3n - 3 

an = 3n - 1 

Find the 25th term of the sequence. 

a25 = 3 × 25 - 1 

= 75 - 1

= 74

Therefore, the 25th term of the sequence is 74. 

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Explicit Formula For Geometric Sequence

The geometric sequence is any sequence where the ratio of any two consecutive terms is the same. The ratio is known as the common ratio (r). The general form of a geometric sequence can be represented as a, ar, ar2, ar3, … arn - 1. For the geometric sequence, the explicit formula is an = arn - 1. 

Where, 

  • a is the first term
  • an is the nth term 
  • r is the common ratio

For example, for the sequence: 1, 2, 4, 8, …, finding the explicit formula

Here, a = 1

r = 2

The explicit formula for geometric sequences is: an = arn - 1
Substituting the value of a and r:

an = 1 × 2n - 1
= 2n - 1

Finding the 5th term:

a5 = 2(5 - 1)

= 24 = 16

So, the 5th term is 16 

Explicit Formula For Harmonic Sequence

A harmonic sequence is a type of sequence where the reciprocals of the terms form an arithmetic sequence. For instance, the harmonic sequence of 2, 4, 6, 8, … is 1/2, 1/4, 1/6, 1/8, …. The general form of a harmonic sequence can be represented as 1/a, 1/(a +d), 1/(a + 2d), …, 1/(a + (n - 1)d). For a harmonic sequence, the explicit formula: an = 1/(a + (n - 1)d)

  • Where a is the first term
  • an is the nth term
  • d is the common difference

For example, find the explicit formula for the harmonic sequence: 1/3, 1/6, 1/9, 1/12. 

We take the reciprocals of the terms: 3, 6, 9, 12, …, to find a and d. 
Here, a = 3

d = 3

an = 1/(a + (n - 1)d)

Substituting the value of a and d

an = 1/(3 + (n - 1)3)

= 1/(3 + 3n - 3)

= 1/(3n)

Finding the 5th term 

an = 1/(3n)

a5 = 1/(3 × 5)

= 1/15

So, the 5th term of the sequence is 1/15

Step-by-Step Use of an Explicit Formula

The use of explicit formula can be understood by the example mentioned below.

Find the 5th term of the sequence where an=3n+2.

Step 1: Identify the formula and substitute n = 5n 

a5 = 3(5)+2

Step 2: Simplify the expression.

a5 = 15 + 2 = 17.

The fifth term of the sequence is 17.

Tips and Tricks to Master Explicit Formulas

Explicit formulas in math are a difficult topic to comprehend, therefore some tips and tricks are useful to master the topic.

Always Identify the Type of Sequence First: Before using any formula, check if it’s arithmetic, geometric, or harmonic.

Write Down the Given Data Clearly: Always list what is given, a1, n, d, or r.

Substitute Carefully: When substituting into the formula, use parentheses to avoid sign or order errors.

Don’t Forget the Order of Operations: Follow BODMAS/PEDMAS rules while simplifying.

Watch for Negative or Fractional Values: If r<0 or d<0 o, terms may alternate signs.

Common Mistakes and How to Avoid Them in Explicit Formulas

The explicit formula is used to find the nth term of a sequence. Students tend to make mistakes when using the explicit formula. Here are some common mistakes and the ways to avoid them.

Real-world Applications of Explicit Formulas

To calculate or predict a specific term in a sequence, we use the explicit formulas. Here are some real-life applications of the explicit formulas.

  • In sports, we use the arithmetic sequence to design a workout routine to plan a steady increase in the repetitions. At any point in training to determine the number of repetitions, we use the explicit formula. 
  • In finance, to calculate the savings, loans, and investments, where the amount forms a geometric sequence, we can use the explicit formula to predict the amount at a particular time. 
  • For linear population growth models, the explicit formula can predict the population at a specific time. 
  • In computer algorithms, we can use explicit formulas to directly calculate the value in the sequence, which helps in improving efficiency. 
  • In environmental studies, explicit formulas can be applied to model patterns such as temperature changes or rainfall distribution over time, helping predict future trends based on previous data.

Problem 1

Find the explicit formula for an arithmetic sequence, where the first term is 5 and the common difference is 3.

Okay, lets begin

an = 3n + 2

Explanation

To find the explicit formula of an arithmetic sequence, we use the formula, an = a + (n - 1)d

Here, a = 5

d = 3

Therefore, an = 5 + (n - 1)3

= 5 + 3n - 3

= 3n + 2

Well explained 👍

Problem 2

If the first term and the common ratio of a geometric sequence are 3 and 2, find the explicit formula.

Okay, lets begin

an = 3 × 2n -1

Explanation

The explicit formula of a geometric sequence is: an = arn - 1

Here, a = 3

r = 2

So, an = 3 × 2n - 1

Well explained 👍

Problem 3

Find the 25th term of a harmonic sequence where the first term is 1/2 and the common difference is 3?

Okay, lets begin

The 25th term is 1/74

Explanation

The explicit formula for a harmonic sequence is: an = 1/(a + (n - 1)d)

Given the first term is 1/2

So, a = 2

d = 3

So, an = 1/(2 + (n - 1)3)

= 1/(2 + 3n - 3)

= 1/(3n -1)

So, the 25th term = a25 = 1/(3 × 25 - 1)

= 1/(75 - 1)

= 1/74

So, the 25th term is 1/74

Well explained 👍

Problem 4

For an arithmetic sequence with a = 3 and d = 5, find the 12th term.

Okay, lets begin

The 12th term is 58

Explanation

If a = 3 and d = 5

The explicit formula of the arithmetic sequence is: an = a + (n - 1)d

a12 = 3 + (12 - 1)5

= 3 + 11 × 5

= 3 + 55 

= 58 

So, here the 12th term is 58

Well explained 👍

Problem 5

What is the common difference of the sequence if the explicit formula is a_n = 7n - 2?

Okay, lets begin

The common difference is 7

Explanation

Here, the explicit formula is: an = 7n - 2

To find the common difference, let’s find the 2nd and 1st terms

a1 = 7 × 1 - 2

= 7 - 2 

= 5

a2 = 7 × 2 - 2

= 14 - 2 

= 12

The difference between any two consecutive terms in a sequence is the common difference.

So, d = a2 - a1

= 12 - 5 

= 7

Well explained 👍

FAQs on Explicit Formulas

1.What is an explicit formula?

The formula used to find the nth term of a sequence when the previous term is unknown is the explicit formula.

2.What is the explicit formula for an arithmetic sequence?

For an arithmetic sequence, the explicit formula is an = a + (n - 1)d, where a is the first term and d is the common difference.

3.What is the explicit formula for a geometric sequence?

The explicit formula for geometric sequences is an = arn - 1, where a is the first term and r is the common difference.

4.What is the explicit formula for a harmonic sequence?

The formula for harmonic sequence is: an = 1/(a + (n - 1)d).

5.What is the explicit formula for 2, 4, 6, 8, ….?

an = a + (n - 1)d is the explicit formula for the arithmetic sequence.  Here a = 2 and d = 2, so an = 2 + (n - 1)2 = 2 + 2n - 2, so an = 2n. 

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.