Arithmetic Sequence Calculator
Find the nth term, sum, and terms of any arithmetic sequence.
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What Is an Arithmetic Sequence?
An arithmetic sequence (or arithmetic progression) is a sequence where the difference between consecutive terms is constant. This constant is called the common difference (d). Example: 3, 7, 11, 15, 19... — the common difference is 4.
Formula for the nth Term
aₙ = a₁ + (n−1)×d, where a₁ is the first term, d is the common difference, and n is the term number. For the sequence 3, 7, 11...: a₁₀ = 3 + (10−1)×4 = 3 + 36 = 39.
Sum of n Terms
The sum of the first n terms is: Sₙ = n/2 × (a₁ + aₙ) = n/2 × (2a₁ + (n−1)×d). For 3, 7, 11..., first 10 terms: S₁₀ = 10/2 × (3+39) = 5 × 42 = 210.
Finding the Common Difference
Given any two terms, d = (aₙ − aₘ) / (n−m). If a₃ = 11 and a₇ = 27, then d = (27−11)/(7−3) = 16/4 = 4.
Real-World Applications
Arithmetic sequences model uniform growth: monthly savings with a fixed addition each month, salary increases by a fixed amount each year, seats in stadium rows increasing by a fixed amount, and depreciation by a fixed annual amount (straight-line depreciation).
Worked Example: Stadium Seating
A concert venue has 20 seats in the first row, and each subsequent row has 4 more seats. How many seats are in row 15, and how many total seats in the first 15 rows?
a₁=20, d=4, n=15. Row 15: a₁₅ = 20 + (15−1)×4 = 20+56 = 76 seats. Total: S₁₅ = 15/2 × (20+76) = 7.5 × 96 = 720 seats.
Arithmetic Sequence Examples at a Glance
| a₁ | d | 5th Term (a₅) | Sum of 5 Terms |
|---|---|---|---|
| 1 | 1 | 5 | 15 |
| 2 | 3 | 14 | 40 |
| 10 | −2 | 2 | 30 |
| 0 | 5 | 20 | 50 |
| 100 | −10 | 60 | 400 |
Frequently Asked Questions
A sequence where each term differs from the previous by a constant value (common difference). Example: 2, 5, 8, 11... (d=3).
aₙ = a₁ + (n−1)×d. For first term 2 and d=3: a₅ = 2 + 4×3 = 14.
Sₙ = n/2 × (a₁ + aₙ) or Sₙ = n/2 × (2a₁ + (n−1)d). The sum of 1 to 100 = 100/2 × (1+100) = 5050.
The sequence is decreasing. Example: 20, 17, 14, 11... (d=−3).
Arithmetic: constant difference (add/subtract). Geometric: constant ratio (multiply/divide). Arithmetic: 2,5,8,11. Geometric: 2,6,18,54.
Carl Gauss reputedly summed 1 to 100 as a child by pairing the first and last terms: 1+100=101, 2+99=101, and so on, giving 50 pairs of 101 = 5050. This generalizes to Sₙ = n/2 × (a₁ + aₙ) — pair the first and last term, multiply by n/2. It works for any arithmetic sequence.
If you know the first term a₁, last term aₙ, and common difference d, the number of terms is n = (aₙ − a₁)/d + 1. For the sequence 5, 8, 11, ..., 41: n = (41−5)/3 + 1 = 36/3 + 1 = 13 terms.