Sequence Term Finder

Math Algebra • Sequences and Series

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

2. Sequence Term Finder

Derive the general term \(a_n\) for an arithmetic or geometric sequence from two arbitrary terms (and optionally a known difference/ratio). Includes consistency checks, steps, term table, and a zoomable partial plot.

Sequence type: Consistency tolerance:

Two known terms

Position \(n_1\):

Value \(a_{n_1}\):

Position \(n_2\):

Value \(a_{n_2}\):

I already know the common difference/ratio

Provided value (d or r):

If provided, the tool will verify it against the two given terms and flag mismatches.

Find a specific term:

Generate term table up to \(N\) (max 200):

Plot first \(N\) terms (max 200):

Connect points Show values on points Auto label density

Ready

Sequence plot

x-axis: n (index) • y-axis: \(a_n\) (term value). Drag to pan • wheel/pinch to zoom • given terms + target term are highlighted.

n: 0, aₙ: 0 sx: 60, sy: 35

Click Calculate to derive \(d\) or \(r\), compute \(a_1\), show \(a_n\), and generate a table.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

Theory: Finding \(a_n\) from Two Terms

A sequence assigns a value \(a_n\) to each index \(n=1,2,3,\dots\). This tool assumes the sequence is either arithmetic (constant difference) or geometric (constant ratio), and reconstructs the general term from two given points \((n_1,a_{n_1})\) and \((n_2,a_{n_2})\).

1) Arithmetic sequences

An arithmetic sequence satisfies \(a_{n}=a_1+(n-1)d\), where \(d\) is the common difference. Using two terms:

\[ d=\frac{a_{n_2}-a_{n_1}}{n_2-n_1}. \]

Once \(d\) is known, back-solve for the first term:

\[ a_1=a_{n_1}-(n_1-1)d. \]

Then the general term is

\[ a_n=a_1+(n-1)d, \]

If the user provides a value of \(d\), the calculator checks whether it predicts both given terms (within a tolerance).

2) Geometric sequences

A geometric sequence satisfies \(a_n=a_1\,r^{\,n-1}\), where \(r\) is the common ratio. Divide the two known terms:

\[ \frac{a_{n_2}}{a_{n_1}}=r^{\,n_2-n_1} \quad\Rightarrow\quad r=\left(\frac{a_{n_2}}{a_{n_1}}\right)^{\frac{1}{n_2-n_1}}. \]

Then recover \(a_1\) using

\[ a_1=\frac{a_{n_1}}{r^{\,n_1-1}}. \]

Real-number warning: if \(\frac{a_{n_2}}{a_{n_1}}<0\) and \(n_2-n_1\) is even, there is no real \(r\). The tool flags this as “no real geometric model.”

3) Consistency checking and edge cases

Consistency: once a model is chosen, the tool predicts both given terms and compares the mismatch to a tolerance.
Same position: if \(n_1=n_2\), two different values cannot both be true; the input is invalid.
Zeros in geometric sequences: if \(a_{n_1}=0\) but \(a_{n_2}\neq 0\), no geometric sequence can satisfy both.

4) Extensions (optional)

More advanced models include arithmetic-geometric forms like \[ a_n=(\alpha+\beta n)\,r^n, \] and “noisy data” fitting where the two points don’t match perfectly (error analysis / best fit).

Frequently Asked Questions

How do you find the common difference d from two terms in an arithmetic sequence?

For two points (n1, a_n1) and (n2, a_n2), the common difference is d = (a_n2 - a_n1) / (n2 - n1). Once d is known, the calculator back-solves a1 and writes a_n = a1 + (n - 1)d.

How do you find the common ratio r from two terms in a geometric sequence?

Using a_n2 / a_n1 = r^(n2 - n1), the ratio is r = (a_n2 / a_n1)^(1 / (n2 - n1)). The calculator then computes a1 = a_n1 / r^(n1 - 1) and forms a_n = a1 x r^(n - 1).

Why does the calculator say there is no real geometric model?

A real r may not exist when a_n2 / a_n1 is negative and the exponent 1/(n2 - n1) requires an even root, or when zeros make the ratio undefined. The tool flags these cases instead of forcing a non-real ratio.

What does the consistency tolerance do in the Sequence Term Finder?

After deriving (or accepting) d or r, the calculator predicts the two given terms and compares any mismatch to the tolerance. This helps detect input mistakes and handles small rounding differences in the provided values.

Rate this calculator

Frequently Asked Questions

Related calculators

Questions related to this topic