Quick answer: In the arithmetic sequence 3, 7, 11, … (d = 4) the 10th term is 3 + 9 × 4 = 39 and the first 10 terms add up to 10 × (3 + 39) ÷ 2 = 210.
How it's calculated
Arithmetic sequences add the same difference d each time: aₙ = a₁ + (n − 1)d, and the sum of the first n terms is Sₙ = n(a₁ + aₙ) ÷ 2.
Geometric sequences multiply by the same ratio r: aₙ = a₁·rⁿ⁻¹ and Sₙ = a₁(1 − rⁿ) ÷ (1 − r), or n·a₁ when r = 1. When −1 < r < 1 the infinite sum converges to a₁ ÷ (1 − r).
In the Fibonacci sequence each term is the sum of the two before it, starting 1, 1, 2, 3, 5, 8. The sum of the first n terms is Fₙ₊₂ − 1. Terms are calculated exactly, so F₁₀₀ shows all 21 digits.
Compound growth is a geometric sequence; for powers on their own, see the exponent calculator.
Worked examples
- Arithmetic a₁ = 3, d = 4, n = 10: a₁₀ = 39, S₁₀ = 210
- Geometric a₁ = 3, r = 2, n = 10: a₁₀ = 3 × 2⁹ = 1,536, S₁₀ = 3 × 1,023 = 3,069
- Fibonacci n = 10 and n = 50: F₁₀ = 55 (sum 143), F₅₀ = 12,586,269,025
Frequently asked questions
How do I tell arithmetic from geometric?
Subtract neighbouring terms: a constant difference means arithmetic. Divide them: a constant ratio means geometric.
Can the difference or ratio be negative?
Yes. A negative d makes the terms fall; a negative r makes them alternate in sign, such as 3, −6, 12, −24.
Where does the Fibonacci sequence start?
This calculator uses F₁ = F₂ = 1 (and F₀ = 0), the usual convention.
What is the 10th term of 3, 7, 11, 15 …?
39. Start at 3 and add 4 nine times. The first ten terms add up to 210.
What does 1 + 1/2 + 1/4 + 1/8 + … add up to?
Exactly 2. Each step covers half the distance that's left, so you get closer and closer to 2 without going past it.
How do I find the common difference or ratio?
For the difference, subtract any term from the next one. In 3, 7, 11, 15 it's 4. For the ratio, divide a term by the one before it.