Hong Life CalculatorsUnderstanding the arithmetic and geometric sequence calculator — concept and principle
A sequence is an ordered list of numbers built by a rule. In an arithmetic sequence each term adds the same common difference d; in a geometric sequence each term multiplies by the same common ratio r. This calculator finds the nth term, the sum of the first n terms, the infinite sum of a geometric series when it exists, and lists the first terms. Arithmetic sequences model steady change; geometric sequences model growth or decay by a constant percentage.
How it is calculated
- Arithmetic nth term: aₙ = a₁ + (n − 1)d.
- Arithmetic sum: Sₙ = n(2a₁ + (n − 1)d) ÷ 2, which equals n(a₁ + aₙ) ÷ 2.
- Geometric nth term: aₙ = a₁ rⁿ⁻¹.
- Geometric sum: Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for r ≠ 1, and Sₙ = n a₁ when r = 1.
- Infinite geometric sum: S = a₁ ÷ (1 − r), only when |r| < 1.
The same input box holds d for arithmetic and r for geometric sequences; the first eight terms are listed.
The principle
To reach the nth term you take n − 1 steps from the first, which is why n − 1 appears in both term formulas. For the arithmetic sum, write the terms forwards and backwards and add them in pairs: every pair totals a₁ + aₙ, and there are n pairs for two copies of the sum, so Sₙ = n(a₁ + aₙ) ÷ 2. For the geometric sum, multiply Sₙ by r and subtract: almost every term cancels, leaving Sₙ(1 − r) = a₁(1 − rⁿ). When |r| < 1, rⁿ shrinks to 0 as n grows, so the sum approaches a₁ ÷ (1 − r).
Worked example
- Inputs: a₁ = 3, d = 2, n = 10, type arithmetic.
- 10th term: a₁₀ = 3 + (10 − 1) × 2 = 3 + 18 = 21.
- Sum: S₁₀ = 10 × (2 × 3 + 9 × 2) ÷ 2 = 10 × 24 ÷ 2 = 120.
- Infinite sum: —, because it applies only to geometric sequences with |r| < 1.
- First terms: 3, 5, 7, 9, 11, 13, 15, 17.
| Term number n | nth term |
|---|---|
| 2.5 | 6 |
| 5 | 11 |
| 7.5 | 16 |
| 10 | 21 |
| 12.5 | 26 |
| 15 | 31 |
| 20 | 41 |
How to read the result
An arithmetic sequence grows along a straight line and its sum grows like n². A geometric sequence with r > 1 grows exponentially and soon overtakes any arithmetic one: with the same inputs but type geometric, the 10th term is 3 × 2⁹ = 1,536 and the sum is 3,069. A ratio between 0 and 1 means decay, and a negative ratio makes the terms alternate in sign.
Common mistakes
- Using n instead of n − 1 in the term formula, which gives the (n + 1)th term.
- Entering a percentage growth as the ratio: 5% growth is r = 1.05, not 5 or 0.05.
- Applying the infinite-sum formula when |r| ≥ 1, where the series diverges.
Frequently asked questions
How do I find the common difference or ratio from two terms?
For an arithmetic sequence, d = (aₘ − aₖ) ÷ (m − k). For a geometric one, r is the (m − k)th root of aₘ ÷ aₖ.
What is the sum 1 + 1/2 + 1/4 + … forever?
It is geometric with a₁ = 1 and r = 1/2, so the infinite sum is 1 ÷ (1 − 1/2) = 2.
Is the Fibonacci sequence arithmetic or geometric?
Neither. Each term is the sum of the two before it, although the ratio of consecutive terms approaches about 1.618.
Sources: Euclid, Elements, Book IX, Proposition 35 (sum of a geometric progression) · James Stewart, Calculus, Chapter 11 (Infinite Sequences and Series)