Find the nth term and the sum of the first n terms of an arithmetic sequence from its first term and common difference.
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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