Arithmetic Sequence Calculator
Find any term. Sum the series. See the pattern behind the numbers.
Sequence & Sum Summary
Exact decimal calculation3 + (10 − 1) × 4
Every term lies on a straight line
Discrete terms • aₙ = 4n − 1
Select a point to inspect its term.
Ten terms. Five equal pairs.
Pair the first term with the last, then work inward. Every pair has the same sum.
For odd n, leave the middle term unpaired. The same sum formula still works.
Formula & worked example
Sₙ = n(a₁ + aₙ) / 2
Find the last term:
a₁₀ = 3 + 9 × 4 = 39
Find the average term:
(3 + 39) / 2 = 21
Multiply by the number of terms:
S₁₀ = 10 × 21 = 210
A sequence lists terms. A series adds them.
Frequently asked questions
Can the common difference be negative?
Yes. A negative difference makes consecutive terms decrease.
Why does the sum formula divide by two?
Writing the series forward and backward gives n pairs, each totaling a₁ + aₙ. This counts the sum twice.
What happens when d = 0 or n = 1?
If d = 0, every term equals a₁ and Sₙ = na₁. If n = 1, both the last term and the sum equal a₁.
Can the number of terms be a decimal?
No. This finite sequence uses a positive integer number of terms. The first term and common difference may be decimals.
How are odd-length sequences paired?
Pair the first and last terms inward, leaving one middle term. Add that middle term once.
Sources & Reference Guides
Independent educational references · No institutional endorsement or expert review claimed.
Find any term. Sum the series. See the pattern behind the numbers.