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101 LearnersLast updated on September 26, 2025
Math Formula for an Arithmetic Sequence Explicit Formula
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
An arithmetic sequence is defined by its common difference and starting term. Let’s learn the explicit formula to calculate any term in an arithmetic sequence.
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.
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.
Mistake 1
Forgetting the First Term
Students sometimes forget to include the first term in the formula, leading to incorrect results. Always remember to start with the first term, a₁, when using the explicit formula.
Mistake 2
Misidentifying the Common Difference
Incorrectly identifying the common difference can lead to errors. Double-check the sequence's terms to ensure the common difference, d, is consistent throughout the sequence.
Mistake 3
Incorrect Term Number
Miscounting the term number, n, is a common mistake. Clearly identify the position of the term you are solving for to avoid this error.
Mistake 4
Confusing Arithmetic and Geometric Sequences
Students often confuse arithmetic sequences with geometric ones. Remember that arithmetic sequences have a constant difference, while geometric sequences have a constant ratio.
Mistake 5
Applying the Wrong Formula
Using the recursive formula instead of the explicit formula can cause confusion. Ensure you are applying the correct formula for the task at hand.
Examples of Problems Using Arithmetic Sequence Explicit Formula
Problem 1
Find the 10th term in the sequence where a₁ = 3 and d = 2.
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
Problem 2
What is the 7th term of an arithmetic sequence where the first term is 5 and the common difference is 4?
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
Problem 3
Calculate the 15th term in the sequence: 2, 5, 8, 11,...
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
Problem 4
What is the 12th term of the sequence with a₁ = 7 and d = -3?
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
Problem 5
Find the 5th term of the arithmetic sequence where the first term is 10 and the common difference is 6.
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
FAQs on Arithmetic Sequence Explicit Formula
1.What is the arithmetic sequence formula?
2.How do you identify an arithmetic sequence?
3.What is the common difference in an arithmetic sequence?
4.How do you find the nth term in an arithmetic sequence?
5.What is a real-life example of an arithmetic sequence?
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.
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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.
