Arithmetic or Geometric Sequence Generator

Math Algebra • Sequences and Series

Written by STEM Calculators Team Published December 19, 2025 Updated February 24, 2026

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1. Arithmetic Or Geometric Sequence Generator

Generate terms, \(a_n\), partial sums \(S_n\), detect AP/GP from data, and visualize as labeled points.

Mode: Up to \(n\) (max 50):

Arithmetic sequence inputs

First term \(a_1\):

Common difference \(d\):

Plot:

Terms \(a_n\) Partial sums \(S_n\) Show point values (on graph)

Ready

Graph

Axes are labeled as n (term index) and value. Drag to pan • wheel/pinch to zoom • auto-fit uses separate x/y scales.

x: 0, y: 0 sx: 60, sy: 35

Click Generate to see formulas, detected parameters (if applicable), and the term table.

Quick quiz (random)

Identify the type and predict the next term.

Click Random sequence to start a quiz.

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Theory: Arithmetic and Geometric Sequences (and Sums)

A sequence is a list of numbers \(a_1, a_2, a_3,\dots\) indexed by \(n=1,2,3,\dots\). A series is the sum of terms, for example \(S_n=a_1+a_2+\cdots+a_n\) (the partial sum).

1) Arithmetic sequences (AP)

In an arithmetic sequence, consecutive terms differ by a constant amount \(d\):

\[ a_{n+1}-a_n=d. \]

The explicit formula is:

\[ \boxed{a_n=a_1+(n-1)d}. \]

The partial sum has a closed form:

\[ \boxed{ S_n=\frac{n}{2}\Big(2a_1+(n-1)d\Big) }. \]

2) Geometric sequences (GP)

In a geometric sequence, consecutive terms are multiplied by a constant factor \(r\):

\[ \frac{a_{n+1}}{a_n}=r \qquad (a_n\neq 0). \]

The explicit formula is:

\[ \boxed{a_n=a_1 r^{\,n-1}}. \]

The partial sum is:

\[ \boxed{ S_n= \begin{cases} n a_1, & r=1\\[6pt] a_1\dfrac{r^n-1}{r-1}, & r\neq 1 \end{cases} }. \]

If \(|r|<1\), then the infinite series converges:

\[ |r|<1 \;\Rightarrow\; \boxed{S_\infty=\frac{a_1}{1-r}}. \]

3) Detecting AP vs GP from data

AP test: check whether all differences \(a_{k}-a_{k-1}\) are (approximately) equal.
GP test: check whether all ratios \(a_{k}/a_{k-1}\) are (approximately) equal (requires \(a_{k-1}\neq 0\)).

Real data may have rounding noise, so the calculator uses a small tolerance when comparing differences/ratios.

4) Arithmetic–geometric extension

A simple mixed model adds an arithmetic part and a geometric part:

\[ \boxed{a_n=a+nd+r^n}. \]

The partial sum splits into three sums:

\[ S_n=\sum_{k=1}^{n}(a+kd+r^k) =na + d\frac{n(n+1)}{2} + \sum_{k=1}^{n}r^k. \]

\[ \sum_{k=1}^{n}r^k= \begin{cases} n, & r=1\\[6pt] \dfrac{r(1-r^n)}{1-r}, & r\neq 1 \end{cases}. \]

In the graph, points are plotted at \((n,a_n)\) (and optionally \((n,S_n)\)). This calculator can display the value of each plotted point directly on the graph.

Frequently Asked Questions

How do I tell if a sequence is arithmetic or geometric?

A sequence is arithmetic if the difference between consecutive terms is constant, and geometric if the ratio of consecutive terms is constant (when the previous term is not zero). The auto-detect option checks these differences or ratios using a tolerance to handle rounding noise.

What formulas does the calculator use for arithmetic sequences?

For an arithmetic sequence, the nth term is a_n = a1 + (n - 1)d. The partial sum is S_n = n/2 x (2a1 + (n - 1)d).

What formulas does the calculator use for geometric sequences and sums?

For a geometric sequence, the nth term is a_n = a1 x r^(n - 1). The partial sum is S_n = a1 x (r^n - 1)/(r - 1) for r != 1 (and S_n = n x a1 for r = 1), and the infinite sum is S_infinity = a1/(1 - r) only when |r| < 1.

What is the arithmetic–geometric mode a_n = a + n d + r^n used for?

It models sequences that combine a linear trend (a + n d) with an exponential component (r^n). The calculator generates its terms and partial sums so you can see how both parts contribute across n.

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