Arithmetic Sequence Calculator – Find Terms & Sum Online
Calculate any term, common difference, or partial sum of an arithmetic sequence with our free online calculator. Enter known values to solve arithmetic progressions instantly.
Understanding Arithmetic Sequences
An arithmetic sequence (or arithmetic progression) is a list of numbers where each term differs from the previous one by a constant amount. That constant is called the common difference.
The Arithmetic Sequence Formula
nth Term: aₙ = a₁ + (n-1)dSum: Sₙ = n/2 × (a₁ + aₙ)Where a₁ is the first term, d is the common difference, and n is the position of the term you want to find.
Worked Examples
Example 1: Basic Sequence
First term a₁ = 2, common difference d = 3, find 10 terms.
a₁₀ = 2 + (10-1) × 3 = 2 + 27 = 29
S₁₀ = 10/2 × (2 + 29) = 5 × 31 = 155Sequence: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29
Example 2: Negative Difference
a₁ = 100, d = -5, find 8 terms.
a₈ = 100 + (8-1) × (-5) = 100 - 35 = 65
S₈ = 8/2 × (100 + 65) = 4 × 165 = 660Sequence: 100, 95, 90, 85, 80, 75, 70, 65
Example 3: Finding the Sum
What's the sum of the first 50 positive integers?
a₁ = 1, d = 1, n = 50
a₅₀ = 1 + 49 × 1 = 50
S₅₀ = 50/2 × (1 + 50) = 25 × 51 = 1275This is the famous formula Gauss supposedly discovered as a schoolboy.
A Quick Fact
The story goes that 7-year-old Carl Friedrich Gauss was told to add all numbers from 1 to 100 as busywork. He instantly replied 5050. He'd realized 1+100=101, 2+99=101, and so on — 50 pairs of 101. That's the arithmetic series formula in action.
Common Questions
What if the common difference is zero?
Then every term equals the first term. It's a constant sequence: 5, 5, 5, 5... The sum is just n times the first term.
Can the common difference be negative?
Absolutely. A negative d means the sequence decreases. These show up in real situations like depreciation or countdown patterns.
How do I find which term equals a specific value?
Rearrange the nth term formula: n = (aₙ - a₁)/d + 1. If n comes out as a whole number, that value exists in the sequence.
What's the difference between a sequence and a series?
A sequence is the list of numbers. A series is the sum of those numbers. This calculator gives you both — the sequence itself and its partial sum.
Where do arithmetic sequences appear in real life?
Staircase steps, seating arrangements in theaters, regular salary increases, depreciation schedules — anywhere something changes by a fixed amount each time.
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