Number Sequence Calculator

Q: How do I tell an AP from a GP?

If the differences between terms are equal it's an AP; if the ratios are equal it's a GP.

Q: Can the terms be fractions?

Yes: type them as decimals (0.5) and the working stays exact.

Q: How big can Fibonacci numbers get here?

They're exact for any n up to 10,000, using whole-number arithmetic.

Find any term and the sum of an arithmetic or geometric progression, insert means, find which term a number is, or sum 1 to n, with exact fractions.

I want to	Arithmetic progression (AP): term and sum
First term (a)	3
Common difference (d)	4
Number of terms (n)	10

I want to

First term (a)

Fractions work too: type them as decimals, like 0.5.

Common difference (d)

Number of terms (n)

For inserting means: how many means to put in between.

a₁₀

Adding 4 each time from 3, term 10 is 39 and the first 10 terms add up to 210.

Exact fraction. Checked exactly.

Method: aₙ = a + (n − 1)d and Sₙ = n/2(2a + (n − 1)d)
Worked out with: AllCalcHub exact fractions
Check: Every step recomputed with exact fractions

Sum of first 10 terms: 210
First terms: 3, 7, 11, 15, 19, 23, 27, 31, …

The first 10 terms

1
nth term: aₙ = a + (n − 1)d
a₁₀ = 3 + 9 × 4 = 39
2
Sum: Sₙ = n/2 × (2a + (n − 1)d)
S₁₀ = 10/2 × (2 × 3 + 9 × 4) = 210
3
Check with Sₙ = n/2 × (first + last)
10/2 × (3 + 39) = 210 ✓

About the Number Sequence Calculator

In an arithmetic progression (AP) you add the same amount each time (3, 7, 11, 15…); in a geometric progression (GP) you multiply by the same ratio (3, 6, 12…). A GP with a ratio between −1 and 1 has a finite sum even with infinitely many terms.

Common follow-up questions ask you to insert arithmetic or geometric means between two numbers, or to use the sums Σk, Σk² and Σk³.

The formula

$a_n = a + (n-1)d, \quad S_n = \frac{n}{2}\big(2a + (n-1)d\big)$

Sₙ (GP): a(rⁿ − 1) ÷ (r − 1)
S∞ (GP): a ÷ (1 − r), when |r| < 1
Σk: n(n + 1)/2
Σk²: n(n + 1)(2n + 1)/6
Σk³: (n(n + 1)/2)²

Worked example

Which term of 3, 8, 13, … is 78?

3 + (n − 1) × 5 = 78
n − 1 = 15, so n = 16

Questions people ask

How do I tell an AP from a GP?

If the differences between terms are equal it's an AP; if the ratios are equal it's a GP.

Can the terms be fractions?

Yes: type them as decimals (0.5) and the working stays exact.

How big can Fibonacci numbers get here?

They're exact for any n up to 10,000, using whole-number arithmetic.