Arithmetic Sequence Calculator
Find the nth term, partial sum, and common difference of any arithmetic sequence. Generate the first n terms.
What Is the Arithmetic Sequence Calculator?
An arithmetic sequence is a list of numbers where each term increases or decreases by a constant amount called the common difference (d). Given any two of the four quantities — first term, common difference, nth term, and partial sum — the other two can be determined. This calculator solves all four and generates up to the first 20 terms.
Formula
How to Use
Enter the first term (a₁) and the common difference (d). Specify how many terms (n) you want to analyse. The calculator shows the nth term, the sum of the first n terms, and a list of all terms. You can also enter the nth term to find the common difference.
Example Calculation
First term = 3, common difference = 7, n = 10: a₁₀ = 3 + 9×7 = 66. S₁₀ = 10/2 × (3 + 66) = 5 × 69 = 345. Sequence: 3, 10, 17, 24, 31, 38, 45, 52, 59, 66.
Understanding Arithmetic Sequence
Arithmetic sequences appear throughout mathematics and everyday life: the natural numbers 1, 2, 3, … are the simplest; equally-spaced marks on a ruler form one; depreciation of an asset by a fixed dollar amount each year produces one; the odd numbers 1, 3, 5, 7, … have d = 2.
The sum formula Sₙ = n(a₁ + aₙ)/2 has a beautiful geometric interpretation: pair the first and last term, then the second and second-to-last, and so on. Each pair sums to the same value (a₁ + aₙ), and there are n/2 such pairs.
Arithmetic sequences are foundational in number theory, combinatorics (arithmetic progressions in primes — the Green-Tao theorem), and computing (loop counters with fixed increments).
Frequently Asked Questions
What is the common difference in an arithmetic sequence?
The common difference d = a₂ − a₁ = a₃ − a₂ = … — the constant amount added to each term to get the next. It can be positive (increasing), negative (decreasing), or zero (constant sequence).
What is the sum formula for an arithmetic series?
The sum of the first n terms is Sₙ = n/2 × (first term + last term) = n/2 × (2a₁ + (n−1)d). This is sometimes called Gauss's formula, as young Gauss is said to have used it to sum 1 through 100.
How is an arithmetic sequence different from a geometric sequence?
In an arithmetic sequence, consecutive terms differ by a constant (addition). In a geometric sequence, they differ by a constant ratio (multiplication). The sum of an infinite arithmetic series diverges; a geometric series may converge if |r| < 1.
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