Sequence & Series Explorer

Enter the first term, common difference or ratio, and number of terms to generate a sequence. See individual terms, partial sums, convergence analysis, and an interactive plot. All calculations run in your browser.

First term (a₁)

Common difference (d)

Number of terms (n)

nth Term

29

Partial Sum

155

Explicit formula

Sum formula

Diverges

n	aₙ	Sₙ
1	2	2
2	5	7
3	8	15
4	11	26
5	14	40
6	17	57
7	20	77
8	23	100
9	26	126
10	29	155

Step-by-Step

Reference Guide

Arithmetic Sequences

In an arithmetic sequence, each term is found by adding a constant value called the common difference. The common difference d is constant between consecutive terms.

General term

a_n = a_1 + (n-1)d

Partial sum

S_n = \frac{n}{2}(2a_1 + (n-1)d)

Geometric Sequences

In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio. The common ratio r is the factor between consecutive terms.

General term

a_n = a_1 \cdot r^{n-1}

Partial sum for

r \ne 1

S_n = a_1 \cdot \frac{1 - r^n}{1 - r}

Convergence of Geometric Series

An infinite geometric series converges when $|r| < 1$ , and its sum is given by the formula below. When $|r| \geq 1$ , the series diverges. Alternating series (negative r) can still converge if $|r| < 1$ .

Infinite sum

S_\infty = \frac{a_1}{1 - r}

Sigma Notation

Sigma notation provides a compact way to write the partial sum of a sequence. The index k runs from the lower bound to the upper bound.

Definition

\sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n

This compact notation represents the partial sum. Each term $a_k$ is evaluated at the corresponding index value and added to the total.