Arithmetic Sequence Calculator — nth Term, Sum & Common Difference
Find any term or sum of an arithmetic sequence. Covers the nth term formula, sum of n terms, common difference, and real-world examples like salary increments.
An arithmetic sequence is the simplest pattern in mathematics: each term is obtained by adding the same fixed amount to the previous one. It shows up in salary schedules, stair design, savings plans, and countless word problems. Calculate any arithmetic sequence with the CalcHub Arithmetic Sequence Calculator.
Key Definitions
First term (a₁): The starting value of the sequence. Common difference (d): The fixed amount added each time. Can be positive, negative, or zero. nth term (aₙ): The value at position n in the sequence.A sequence is arithmetic if and only if the difference between consecutive terms is constant:
aₙ₊₁ − aₙ = d for all n
The nth Term Formula
$$a_n = a_1 + (n-1)d$$
Example: Find the 15th term of 3, 7, 11, 15, ...Here a₁ = 3 and d = 4 (each term is 4 more than the previous).
a₁₅ = 3 + (15 − 1)×4 = 3 + 56 = 59
The Sum Formula
The sum of the first n terms:
$$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$$
Where does this come from? Gauss's trick: pair the first and last terms, second and second-to-last, etc. Each pair sums to a₁ + aₙ, and there are n/2 such pairs. Example: Sum of first 15 terms of 3, 7, 11, ...Method 1: S₁₅ = (15/2)(3 + 59) = (15/2)(62) = 15 × 31 = 465
Method 2: S₁₅ = (15/2)[2×3 + 14×4] = (15/2)[6 + 56] = (15/2)(62) = 465 ✓
Real-World Example: Salary Increments
You join a company at ₹40,000/month with a guaranteed ₹3,000 annual increment.
| Year | Salary (₹) |
|---|---|
| 1 | 40,000 |
| 2 | 43,000 |
| 3 | 46,000 |
| 10 | 67,000 |
| 20 | 97,000 |
Total earnings over 10 years: S₁₀ = (10/2)(40,000 + 67,000) = 5 × 107,000 = ₹535,000/year average × 10 = ₹5,35,000 annual
(This shows average salary × 10 years for annual totals.)
Finding d and a₁ from Two Terms
If you know aₘ and aₙ:
$$d = \frac{a_n - a_m}{n - m}$$
Example: The 5th term is 23 and the 10th term is 43. d = (43 − 23)/(10 − 5) = 20/5 = 4 a₁ = a₅ − 4×4 = 23 − 16 = 7 Sequence: 7, 11, 15, 19, 23, 27, 31, 35, 39, 43 ✓Arithmetic vs. Geometric
| Property | Arithmetic | Geometric |
|---|---|---|
| Pattern | Add constant d | Multiply constant r |
| nth term | a₁ + (n−1)d | a₁ × rⁿ⁻¹ |
| Sum (finite) | n/2·(a₁+aₙ) | a₁(1−rⁿ)/(1−r) |
| Graph shape | Straight line | Exponential curve |
Can a sequence have d = 0?
Yes — this is a constant sequence (e.g., 5, 5, 5, 5...). Every term equals a₁, and the sum is simply n × a₁. Not very exciting, but technically arithmetic.
How are arithmetic sequences related to linear functions?
Very directly. The nth term formula aₙ = a₁ + (n−1)d can be rewritten as aₙ = dn + (a₁ − d). This is a linear function of n, with slope d and y-intercept (a₁ − d). Plotting (n, aₙ) gives a straight line — hence the connection to linear equations.
What's the sum of integers from 1 to 100?
This is the sum of the arithmetic sequence 1, 2, 3, ..., 100 (a₁=1, d=1, n=100).
S₁₀₀ = (100/2)(1 + 100) = 50 × 101 = 5,050
This is the famous result Gauss reportedly calculated as a schoolboy.