Arithmetic Sequence Calculator
Calculate nth term, sum, and generate arithmetic sequences with common difference.
Sequence Parameters
Sequence Terms
Formulas Used
nth Term Formula
an = a1 + (n - 1) x d
Sum Formula
Sn = (n/2) x (a1 + an)
Term #10 (a10)
29
Step-by-Step Calculation
Given values
a1 = 2, d = 3, n = 10
Calculate nth term
a10 = 2 + (10 - 1) x 3 = 29
Calculate sum
S10 = (10/2) x (2 + 29) = 155
About Arithmetic Sequences
Definition
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant is called the common difference (d).
Examples
- 2, 5, 8, 11, 14... (d = 3)
- 10, 7, 4, 1, -2... (d = -3)
- 1, 1.5, 2, 2.5, 3... (d = 0.5)
What Is an Arithmetic Sequence Calculator?
An arithmetic sequence (also called an arithmetic progression) is a list of numbers where the difference between any two consecutive terms is constant. That constant is the common difference d. For example, 2, 5, 8, 11, 14... has d = 3 because each term adds 3 to the previous one. This calculator takes the first term a₁, the common difference d, and the number of terms n, and computes the nth term, the sum of the first n terms, and the average — then displays the actual sequence.
Arithmetic sequences model any situation with uniform growth or decline: saving a fixed amount each month, increasing production by a constant number of units per week, or decreasing temperature by a steady rate. The formulas are simple enough to use by hand, but the calculator saves you from arithmetic errors and lets you explore how changing a single parameter affects the entire sequence.
Arithmetic Sequence Formulas
Two core formulas power this calculator. The nth term formula computes any term directly without generating all preceding terms. The sum formula uses the first and nth terms for efficiency — no need to add every term individually.
Arithmetic Sequence Formulas
Where:
- a₁= The first term of the sequence
- d= The common difference — the constant amount added (or subtracted if negative) to each term to get the next
- n= The position in the sequence — which term number you want to compute
Understanding the Results
| Output | Explanation |
|---|---|
| nth Term (aₙ) | The value at position n: aₙ = a₁ + (n − 1) × d. For n = 10, a₁ = 2, d = 3: a₁₀ = 2 + 9 × 3 = 29. |
| Sum (Sₙ) | Total of the first n terms: Sₙ = (n/2) × (a₁ + aₙ). For the example above: S₁₀ = (10/2) × (2 + 29) = 5 × 31 = 155. |
| Average | Mean of the first n terms: average = Sₙ / n. In an arithmetic sequence, this equals (a₁ + aₙ)/2 — the midpoint of the first and last term. |
The sequence display shows all terms up to n (capped at 20 for readability), with the nth term highlighted. This visual confirmation helps you spot whether the sequence is increasing (d > 0), decreasing (d < 0), or constant (d = 0).
How to Use This Calculator
- Enter first term (a₁): Type the starting value. Can be any number — positive, negative, integer, or decimal.
- Enter common difference (d): The amount each term increases by. Use positive for growth (2, 5, 8...), negative for decline (10, 7, 4...), or zero for constant sequence (5, 5, 5...).
- Enter number of terms (n): Which term position you want to find. Must be a positive integer. The calculator generates up to min(n, 20) terms for display.
- Review the results: The nth term value is highlighted. Below it, result cards show the nth term, sum, average, and common difference. The step-by-step panel walks through both formulas with your specific numbers plugged in.
Real-World Applications
Arithmetic sequences model linear growth in countless real-world contexts. In personal finance, if you save $200 per month with no interest, your total savings after n months follow an arithmetic sequence with d = 200. The sum formula tells you exactly how much you'll have after any number of months. In manufacturing, if a factory increases production by 50 units per week, the weekly output is arithmetic, and the sum gives total units produced over a period.
In construction, the rows of an amphitheater or stadium seating are often arithmetic — each row has a fixed number more seats than the row in front. The sum formula gives total seating capacity. In education, arithmetic sequences are the first type of sequence students encounter, bridging basic arithmetic to algebra and preparing the ground for more complex patterns like geometric sequences and series.
Worked Examples
Standard Arithmetic Sequence
Problem:
Find the 10th term and sum of an arithmetic sequence with a₁ = 2 and d = 3.
Solution Steps:
- 1Enter a₁ = 2, d = 3, n = 10.
- 2Nth term: a₁₀ = 2 + (10 - 1) × 3 = 2 + 27 = 29.
- 3Sum: S₁₀ = (10/2) × (2 + 29) = 5 × 31 = 155.
- 4Average: 155/10 = 15.5 = (2 + 29)/2.
Result:
a₁₀ = 29, Sum = 155, Average = 15.5. Sequence: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29.
Declining Sequence (Negative d)
Problem:
A sequence starts at 100 and decreases by 7 each term. What is the 15th term?
Solution Steps:
- 1Enter a₁ = 100, d = -7, n = 15.
- 2a₁₅ = 100 + (15 - 1) × (-7) = 100 + 14 × (-7) = 100 - 98 = 2.
- 3Sum: S₁₅ = (15/2) × (100 + 2) = 7.5 × 102 = 765.
- 4Average: 765/15 = 51.
Result:
a₁₅ = 2, Sum = 765. The sequence drops from 100 down to 2 over 15 terms.
Uniform Savings Plan
Problem:
You save $200 the first month and increase savings by $50 each month. How much do you save in month 12, and what's the total after 12 months?
Solution Steps:
- 1Enter a₁ = 200, d = 50, n = 12.
- 2a₁₂ = 200 + (12 - 1) × 50 = 200 + 550 = 750.
- 3Sum: S₁₂ = (12/2) × (200 + 750) = 6 × 950 = 5,700.
Result:
Month 12 savings: $750. Total over 12 months: $5,700. Average monthly savings: $475.
Tips & Best Practices
- ✓The sum formula Sₙ = (n/2) × (a₁ + aₙ) is the most efficient way to sum an arithmetic sequence — no need to add terms one by one.
- ✓For a sequence with an odd number of terms, the average equals the middle term exactly.
- ✓If d = 0, every term equals a₁ and the sum is simply n × a₁.
- ✓The sequence display caps at 20 terms to avoid cluttering — use the nth term formula for positions beyond 20.
- ✓You can verify the sum manually for small n by adding the displayed terms — it should match the calculator's sum exactly.
- ✓The average of an arithmetic sequence always equals (a₁ + aₙ)/2, regardless of n or d.
Frequently Asked Questions
Sources & References
Last updated: 2026-06-06
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Editorial Note
MyCalcBuddy Editorial Team
This page is maintained as an educational calculator reference.
Formula Source: Handbook of Mathematical Functions
by Abramowitz & Stegun