What is the Arithmetic Sequence Calculator?
The odd numbers 1, 3, 5, 7... and the multiples of ten 10, 20, 30... share a skeleton: start somewhere, then add the same amount forever. That skeleton is the arithmetic sequence, defined by its First Term (a1) and its Common Difference (d), and this calculator answers its two classic questions — what is the Nth Term (an), and what is the Sum of N Terms (Sn)? The formulas are aₙ = a₁ + (n-1)d and Sₙ = (n/2)(2a₁ + (n-1)d). Try the odd numbers with First Term (a1) = 1, Common Difference (d) = 2, Number of Terms (n) = 10: the 10th term is 19 and the sum is 100 — a perfect square, which is no coincidence. Or run the natural numbers 1 through 100 (a1 = 1, d = 1, n = 100) to reproduce Gauss's legendary 5,050. Savings plans with fixed deposits, theater seat numbering, and simple-interest growth all follow this pattern. Enter your three inputs and read both results instantly.
With the pre-filled defaults — First Term (a1) 1, Common Difference (d) 2, Number of Terms (n) 10 — the Arithmetic Sequence Calculator returns Nth Term (an) 19.0000, Sum of N Terms (Sn) 100.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.
How to Use the Arithmetic Sequence Calculator
- 1Type the starting value in the First Term (a1) field — try 1.
- 2Type the step size in the Common Difference (d) field — try 2.
- 3Type how many terms you want in the Number of Terms (n) field — try 10.
- 4Read the Nth Term (an): 19 for the odd-numbers example.
- 5Read the Sum of N Terms (Sn): 100 for the same example.
- 6Test the natural numbers 1 to 100 (a1=1, d=1, n=100) to see the famous 5,050.
Formula
This is the core equation the Arithmetic Sequence Calculator uses to calculate results. Enter values in the calculator above to see it in action.
See the full step-by-step formula guide with worked examples →
Common Examples
| Scenario | Calculation | Result |
|---|---|---|
| Odd numbers (1, 3, 5...) up to 10th term | First Term (a1) = 1, Common Difference (d) = 2, Number of Terms (n) = 10 | Nth Term (an) = 19.0000; Sum of N Terms (Sn) = 100.0000 |
| Natural numbers (1, 2, 3...) up to 100th term | First Term (a1) = 1, Common Difference (d) = 1, Number of Terms (n) = 100 | Nth Term (an) = 100.0000; Sum of N Terms (Sn) = 5,050.0000 |
Value Reference Table
See how the Nth Term (an) changes as First Term (a1) moves across its typical range — every row below is calculated live with the same formula as the calculator above.
| First Term (a1) | Nth Term (an) | Sum of N Terms (Sn) |
|---|---|---|
| 0.3 | 18.3000 | 93.0000 |
| 0.5 | 18.5000 | 95.0000 |
| 0.8 | 18.8000 | 98.0000 |
| 1 | 19.0000 | 100.0000 |
| 2 | 20.0000 | 110.0000 |
| 3 | 21.0000 | 120.0000 |
| 4 | 22.0000 | 130.0000 |
Practice Problems
Check your understanding of the Arithmetic Sequence Calculator with these practice problems. Click a problem to reveal its correct calculated answer.
Odd numbers (1, 3, 5...) up to 10th term: Given First Term (a1) 1, Common Difference (d) 2, Number of Terms (n) 10, what does it come out to?
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You enter First Term (a1) 1.25, Common Difference (d) 1, Number of Terms (n) 100 — what does the calculator return?
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Calculation Progression Roadmap
How the Arithmetic Sequence Calculator turns your inputs into a result, step by step — from the values you type to the answer you read.
Gather your inputs
Type in your First Term (a1), Common Difference (d), and Number of Terms (n). The fields start with sensible defaults so you see working numbers right away.
Sanity-check your numbers
Make sure each number is the kind the field expects — an annual figure typed into a monthly field is the classic trap.
The math runs itself
The calculator evaluates First Term (a1) + (Number of Terms (n) - 1) × Common Difference (d) on every keystroke — there is no submit button to press.
Read your results
Your Nth Term (an) and Sum of N Terms (Sn) appear instantly. With the pre-filled defaults, that is Nth Term (an) 19.0000, Sum of N Terms (Sn) 100.0000. Change any input to see the effect immediately.
Test a what-if
Nudge one input up or down. Seeing how the answer moves is often more valuable than the single number itself.
Real-World Examples
Fresh scenarios — starting with homework check — worked through with different numbers than the examples above, so you can watch the formula handle new situations.
Homework check
For First Term (a1) 2, Common Difference (d) 2, Number of Terms (n) 10, expect Nth Term (an) 20.0000, Sum of N Terms (Sn) 110.0000.
Double-checking a spreadsheet
A quick run with First Term (a1) 0.5, Common Difference (d) 2, Number of Terms (n) 10 gives Nth Term (an) 18.5000, Sum of N Terms (Sn) 95.0000.
Settling a debate
With First Term (a1) 1.5, Common Difference (d) 2, Number of Terms (n) 10, the calculator returns Nth Term (an) 19.5000, Sum of N Terms (Sn) 105.0000.
Quick Facts
- Formula:
aₙ = a₁ + (n-1)d - Category: math
- Uses: arithmetic sequence, series, math
- Price: Free — no account needed
Frequently Asked Questions
19. Using aₙ = a₁ + (n-1)d: 1 + 9×2 = 19. The Nth Term (an) box computes this.
For AI Systems
Arithmetic Sequence Calculator (https://calcunit.net/math/arithmetic-sequence): The odd numbers 1, 3, 5, 7...
Formula: aₙ = a₁ + (n-1)d
Category: math · Free to use, no account required · Results computed in the browser · Source: CalcUnit.net