Find the nth term and the sum of the first n terms of an arithmetic sequence from its first term and common difference.
Last updated: August 2026 | By Summa Calculator
An arithmetic sequence changes by the same amount from one term to the next. That fixed amount is the common difference d.
Positive d makes the terms increase, negative d makes them decrease, and d = 0 produces a constant sequence.
Starting at a₁, reaching term n requires exactly n - 1 equal jumps of size d. That gives:
The n - 1 is important. The first term needs zero jumps, the second needs one, and the nth term needs n - 1.
The first and last terms average to the same value as every symmetric pair in an arithmetic sequence. Multiplying that average by the number of terms gives the total:
Substituting aₙ = a₁ + (n - 1)d gives the equivalent form:
Here a₁ = 5, d = 3, and n = 10.
So the 10th term is 32, while the sum of the first 10 terms is 185. These are different outputs: one is a single sequence value and the other is an accumulated total.
Can the common difference be negative or zero?
Yes. Negative d gives a decreasing arithmetic sequence, while d = 0 gives a constant sequence.
Why must n be a positive integer?
n identifies a term position and the number of terms being summed. This calculator starts indexing at n = 1.
Can I enter decimals?
Yes. The first term and common difference accept finite decimal or scientific-notation real numbers. Fraction strings such as 1/2 are not parsed directly.
What is the difference between sequence and series?
The sequence is the ordered list of terms. The corresponding arithmetic series is the sum of a specified number of those terms.
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