Arithmetic Sequence Calculator
An arithmetic sequence adds the same fixed amount at every step. This finds any term directly, and the sum of the first n terms, without listing out every value in between.
Inputs
Result
aₙ = 98
Sum of first 20 terms = 1,010
How the arithmetic sequence calculator works
The nth term: aₙ = a + (n−1)d, where a is the first term and d is the common difference.
The sum of the first n terms: Sₙ = (n/2) × (a + aₙ) — the average of the first and last term, times how many terms there are.
Worked example: first term 3, common difference 5, 20th term
- a₂₀ = 3 + (20−1)×5 = 3 + 95 = 98.
- Sum of first 20 terms = (20/2) × (3 + 98) = 10 × 101 = 1,010.
Common mistakes to avoid
Using (n) instead of (n−1) in the term formula
The first term itself, a₁, adds zero multiples of d — it's a + (1−1)d = a. Forgetting the −1 shifts every term calculation by exactly one step of d.
Assuming a negative common difference means an error
A negative d simply means a decreasing sequence — perfectly valid, and the sum formula works identically regardless of the sign of d.
Frequently asked questions
What real situations follow an arithmetic sequence?
Anything that increases or decreases by a fixed amount each step — simple (non-compounding) savings deposits, seating rows that each add the same number of seats, or a countdown timer.
How is this different from a geometric sequence?
Arithmetic adds a fixed amount each step; geometric multiplies by a fixed ratio each step — the two grow completely differently over many terms, linear versus exponential.
Can I find which term number has a specific value?
Rearrange the formula: n = (aₙ − a)/d + 1. This tool computes forward from n, but the same formula runs in reverse by hand.
Why does the sum formula use the average of first and last term?
Because in an arithmetic sequence, pairing the first and last, second and second-last, and so on, always gives the same sum — the classic trick often attributed to a young Gauss adding 1 to 100.
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