Math Formula for an Arithmetic Sequence Explicit Formula
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Last updated on September 26, 2025

In mathematics, an arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference. In this topic, we will learn about the explicit formula for arithmetic sequences and how to use it to find terms in the sequence.

List of Math Formulas for Arithmetic Sequence Explicit Formula

Math Formula for Arithmetic Sequence

The explicit formula for an arithmetic sequence allows us to find any term in the sequence without knowing the previous term.

It is calculated using the formula: aₙ = a₁ + (n - 1) * d where aₙ is the nth term, a₁ is the first term, n is the term number, and d is the common difference.

Example Problems Using Arithmetic Sequence Explicit Formula

To solidify understanding, let's look at some examples of how to use the arithmetic sequence explicit formula.

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Tips and Tricks to Memorize Arithmetic Sequence Formula

Students may find math formulas challenging, but here are some tips to master the arithmetic sequence formula.

Visualize the sequence as a linear graph with the slope representing the common difference.

Practice deriving the formula by starting with simple sequences.

Use mnemonic devices to remember that the formula involves the first term and the common difference.

Real-Life Applications of Arithmetic Sequence Formula

In real life, arithmetic sequences appear in various contexts. Here are some applications of the arithmetic sequence formula.

In finance, calculating equal installment payments over time involves arithmetic sequences.

In construction, determining the number of steps or rows in evenly spaced designs uses arithmetic sequences.

In daily planning, predicting future events with a regular schedule can be modeled with arithmetic sequences.

Common Mistakes and How to Avoid Them While Using Arithmetic Sequence Formula

Students make errors when using the arithmetic sequence formula. Here are some mistakes and ways to avoid them, to understand it fully.

Problem 1

Find the 10th term in the sequence where a₁ = 3 and d = 2.

Okay, lets begin

The 10th term is 21.

Explanation

Using the formula aₙ = a₁ + (n - 1) * d, we substitute:

a₁ = 3, n = 10, d = 2

a₁₀ = 3 + (10 - 1) * 2

a₁₀ = 3 + 18 a₁₀ = 21

Well explained 👍

Problem 2

What is the 7th term of an arithmetic sequence where the first term is 5 and the common difference is 4?

Okay, lets begin

The 7th term is 29.

Explanation

Using the formula aₙ = a₁ + (n - 1) * d, we substitute:

a₁ = 5, n = 7, d = 4

a₇ = 5 + (7 - 1) * 4

a₇ = 5 + 24 a₇ = 29

Well explained 👍

Problem 3

Calculate the 15th term in the sequence: 2, 5, 8, 11,...

Okay, lets begin

The 15th term is 44.

Explanation

First, find the common difference: d = 5 - 2 = 3.

Using the formula aₙ = a₁ + (n - 1) * d, we substitute:

a₁ = 2, n = 15, d = 3

a₁₅ = 2 + (15 - 1) * 3

a₁₅ = 2 + 42 a₁₅ = 44

Well explained 👍

Problem 4

What is the 12th term of the sequence with a₁ = 7 and d = -3?

Okay, lets begin

The 12th term is -26.

Explanation

Using the formula aₙ = a₁ + (n - 1) * d, we substitute:

a₁ = 7, n = 12, d = -3

a₁₂ = 7 + (12 - 1) * (-3)

a₁₂ = 7 - 33 a₁₂ = -26

Well explained 👍

Problem 5

Find the 5th term of the arithmetic sequence where the first term is 10 and the common difference is 6.

Okay, lets begin

The 5th term is 34.

Explanation

Using the formula aₙ = a₁ + (n - 1) * d, we substitute:

a₁ = 10, n = 5, d = 6

a₅ = 10 + (5 - 1) * 6

a₅ = 10 + 24 a₅ = 34

Well explained 👍

FAQs on Arithmetic Sequence Explicit Formula

1.What is the arithmetic sequence formula?

The formula to find any term in an arithmetic sequence is: aₙ = a₁ + (n - 1) * d

2.How do you identify an arithmetic sequence?

An arithmetic sequence is identified by a constant difference between consecutive terms.

3.What is the common difference in an arithmetic sequence?

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

4.How do you find the nth term in an arithmetic sequence?

Use the explicit formula aₙ = a₁ + (n - 1) * d, where a₁ is the first term, n is the term number, and d is the common difference.

5.What is a real-life example of an arithmetic sequence?

A real-life example of an arithmetic sequence is calculating equal payments for a loan over time.

Glossary for Arithmetic Sequence Explicit Formula

  • Arithmetic Sequence: A sequence of numbers with a constant difference between consecutive terms.
  • Explicit Formula: A formula that allows direct computation of any term in a sequence.
  • Common Difference: The consistent difference between terms in an arithmetic sequence.
  • Term: An individual element or number in a sequence.
  • Constant: A value that does not change.

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.