The Arithmetic Sequence Calculator computes the nth term, common difference, and sum of terms for any arithmetic sequence. Enter the values you know, and it returns the missing results using the standard arithmetic sequence formulas.
Arithmetic sequences show up in math, finance, and data patterns. Once you know the first term and the step size, you can predict future terms and total sums quickly and accurately.
What Is an Arithmetic Sequence?
An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. That constant difference is called the common difference.
Examples include sequences like 3, 7, 11, 15, … where each term increases by 4.
Key Variables (Use These in the Calculator)
- a1: the first term of the sequence (term number 1).
- d: the common difference (how much the sequence changes each step).
- n: the term index you want (for example, n = 10 means the 10th term).
- an: the nth term (the result for term number n).
- sum: the sum of the first n terms (often written as Sₙ).
Core Formulas You’ll Use
Arithmetic sequence formulas are simple and reliable. The calculator uses these exact relationships to compute results from your inputs.
1) nth Term Formula
The nth term tells you the value at position n:
an = a1 + (n − 1) × d
2) Common Difference Formula
If you know the first term, the nth term, and n, you can solve for d:
d = (an − a1) / (n − 1)
This works when n > 1. When n = 1, the sequence has only one term, so d cannot be determined from that information alone.
3) Sum of First n Terms
The sum of the first n terms is:
Sₙ = n × (a1 + an) / 2
Because an depends on a1, d, and n, the calculator can compute the sum even if you don’t directly enter an.
How to Use the Arithmetic Sequence Calculator
To get accurate results, choose the set of inputs that matches what you want to find.
- Find the nth term (an): enter a1, d, and n.
- Find the common difference (d): enter a1, an, and n (with n > 1).
- Find the sum (Sₙ): enter a1, d, and n (or provide an if your calculator mode supports it).
If you enter contradictory values, the calculator will prioritize the formulas it can compute from the available fields and will show errors for invalid inputs.
Worked Example 1: Predict the 12th Term
Suppose a sequence starts at a1 = 5 and increases by d = 2 each step. You want n = 12.
Use the nth term formula:
an = 5 + (12 − 1) × 2 = 5 + 11 × 2 = 27
So the 12th term is 27.
Worked Example 2: Total Up the First 20 Terms
Now take a1 = 10 and d = 3. You want the sum of the first n = 20 terms.
First compute the 20th term:
an = 10 + (20 − 1) × 3 = 10 + 19 × 3 = 67
Then compute the sum:
Sₙ = 20 × (10 + 67) / 2 = 20 × 77 / 2 = 770
The sum of the first 20 terms is 770.
Common Real-Life Use Cases
Arithmetic sequences are everywhere when a quantity changes by a constant amount each step.
- Budgeting and saving: adding the same amount each month.
- Manufacturing or production: consistent increments in measurements or outputs.
- School and training: steady increases like points per lesson.
Units and Conversions (Important)
Arithmetic sequence formulas work with numbers, but units matter. If your terms represent money, time, or distance, keep units consistent across a1, d, and an.
If you switch units (for example, meters to centimeters), convert inputs first. The calculator focuses on arithmetic sequence math; it assumes all numeric inputs are in compatible units.
Frequently Asked Questions
How do I find the nth term of an arithmetic sequence?
Use an = a1 + (n − 1) × d. Enter the first term a1, the common difference d, and the term index n. If d is negative, the sequence decreases each step. The calculator returns an directly from these values.
What if I know the first term and the nth term but not the common difference?
Rearrange the formula: d = (an − a1) / (n − 1). This requires n to be greater than 1. If n equals 1, you only have one term, so the common difference cannot be determined from that data.
How do I calculate the sum of the first n terms?
Compute Sₙ = n × (a1 + an) / 2. If you don’t know an, the calculator finds an using an = a1 + (n − 1) × d and then plugs it into the sum formula. This gives the total.
Can an arithmetic sequence have a negative common difference?
Yes. A negative d means each term is smaller than the previous one by a constant amount. The formulas still work the same way. Just be consistent: use negative values for d and keep n as a positive integer.
What inputs are valid for the calculator?
You must enter numeric values for a1, d (or an), and n, depending on what you’re solving for. The term index n must be an integer greater than or equal to 1. The calculator flags invalid values and avoids divide-by-zero errors.
Bottom Line
An Arithmetic Sequence Calculator makes arithmetic sequence problems fast and dependable. With the correct inputs, you can compute the nth term, common difference, and sum of terms using standard formulas.
Use it for homework, planning, and real-world patterns where the change per step stays constant.
