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Arithmetic Sequence Formula
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. Knowing the arithmetic sequence formula allows you to find any term in the sequence (even the 1,000th term!) and calculate the sum of those numbers without adding them one by one. For quick computations, use the arithmetic sequence calculator or the summation calculator to analyze sequences and series instantly.
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What is an arithmetic sequence?
An arithmetic sequence is a progression of numbers where you add (or subtract) the same number to get from one term to the next. That added value is called the common difference, represented by the variable d.
For example, in the sequence: 5, 8, 11, 14, 17..., the common difference is 3. Each term is found by adding 3 to the previous term.
The arithmetic sequence formulas
There are two core formulas used to solve arithmetic sequence problems: one to find a specific term, and one to find the sum of terms.
Formula 1: The Nth Term Formula
a_n = a_1 + (n − 1)d
Formula 2: The Sum of Terms (Series) Formula
S_n = (n / 2) × (a_1 + a_n)
Alternative version: S_n = (n / 2) × [2a_1 + (n − 1)d]
The variables
- a_n = the nth term in the sequence (the value of the target term)
- a_1 = the first term in the sequence
- n = the term position number (e.g., n = 10 for the 10th term)
- d = the common difference between consecutive terms
- S_n = the sum of the first n terms
Worked Example 1: Finding a Specific Term
Let's find the 30th term of the arithmetic sequence: 4, 9, 14, 19, 24...
- Identify the first term: a_1 = 4.
- Calculate the common difference: d = 9 − 4 = 5.
- Identify the term position we want: n = 30.
- Set up the nth term equation: a_30 = 4 + (30 − 1) × 5.
- Simplify and calculate:
- a_30 = 4 + (29) × 5
- a_30 = 4 + 145
- a_30 = 149
The 30th term of this sequence is 149.
Worked Example 2: Finding the Sum of Terms
Let's calculate the sum of the first 30 terms of the same sequence (4, 9, 14, 19... up to 149).
- Identify the first term: a_1 = 4.
- Identify the 30th term (calculated above): a_30 = 149.
- Identify the number of terms: n = 30.
- Set up the sum equation: S_30 = (30 / 2) × (4 + 149).
- Simplify and calculate:
- S_30 = 15 × (153)
- S_30 = 2295
The sum of the first 30 terms is 2,295.
Common Arithmetic Sequence Mistakes
- Confusing n and a_n. The variable n represents the index or placement position (such as term #15), whereas a_n represents the actual numerical value at that position (such as 79). Make sure you don't swap these variables.
- Incorrect sign for negative differences. If a sequence goes down (e.g., 20, 17, 14...), the common difference is negative (d = −3). If you plug in a positive number, the sequence terms will incorrectly grow larger.
- Order of operations error. When using a_1 + (n − 1)d, you must calculate the subtraction inside parentheses, multiply by d, and then add a_1. Do not add a_1 to n before multiplying.
Run the numbers
Solve sequences and sum series instantly with our algebra calculators:
Arithmetic Sequence Calculator
Generate formulas, sequence terms, and find any nth term of an arithmetic progression.
Summation Calculator
Add up terms of a series using standard sigma notation with step-by-step math.
Scientific Calculator
Evaluate complex sequences, fractions, and exponential formulas easily.
Frequently asked questions
An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference (d).
You can find the common difference by subtracting any term in the sequence from the term that follows it: d = a_n - a_(n-1). For example, in the sequence 3, 7, 11, 15, the common difference is 7 - 3 = 4.
In an arithmetic sequence, terms are found by adding or subtracting a constant difference (d) to the previous term. In a geometric sequence, terms are found by multiplying the previous term by a constant ratio (r).
The sum of the first n terms of an arithmetic sequence (also called an arithmetic series) is calculated using: S_n = (n / 2) × (a_1 + a_n), where a_1 is the first term and a_n is the nth term.
The calculator allows you to enter a few known parts of a sequence (like the first term and common difference) and instantly generates the nth term, the sum of terms, and the list of numbers.