Geometric sequence

Calculates geometric progression terms from n to p.
Calculates the sum of terms from n to p.
You can enter numbers or fractions.
p ≥ n

Use this tool to compute the terms and the sum of terms for a geometric progression defined by its common ratio and the first term.

Geometric sequence

A geometric sequence is a sequence of non-zero numbers in which each term is obtained by multiplying the previous term by a fixed value called the common ratio.

The formula for the common ratio is :

`d = U_(n+1) / U_n`

`U_(n+1)` is the (n+1)-th term of the sequence.

`U_n` is the n-th term of the sequence.

Examples of geometric sequence
The sequence of numbers 1, 10, 100, 1000, 10000, 100000... is a geometric sequence of first term 1 and common ratio 10,

`U_1 = 1`
For all n > 1, `U_n = 10 * U_(n-1)`

Geometric sequence general term
The following formula calculates the general term (or n-th term) of a geometric sequence with first term `U_1` and common ratio r, for every integer n >= 1,

`U_n = U_1 * r^(n-1)`

From that, we can deduce another formula that relates any two terms of the sequence. So for every two non-zero integers n and p,

`U_n = U_p * r^(n-p)`

Sum of consecutive terms
The sum of the numbers of a geometric sequence is called a geometric series, we calculate it as follows,

`S_n = U_1 + U_2 + U_3... + U_n`

`S_n = U_1 + r*U_1 + r^2*U_1+....+ r^(n-1)*U_1`

`S_n = U_1 * (1 + r + r^2 + ... + r^(n-1))`

`S_n = U_1 * \frac{1-r^n}{1-r}`, assuming that `(r!=1)`

See also

Arithmetic sequence