Explicit Formula Calculator

What is Explicit Formula Calculator

An Explicit Formula Calculator is a tool that takes a sequence definition—often recursive or described by a rule—and returns the closed-form expression (the explicit formula) for the nth term. Rather than computing terms one-by-one, the explicit formula gives a direct relationship \(a_n = f(n)\) so you can compute any term efficiently. This is especially helpful in algebra, discrete mathematics, computer science, and any situation where quick access to arbitrary-indexed terms is needed.

About the Explicit Formula Calculator

The calculator supports common sequence families (arithmetic, geometric), simple linear recurrences, and typical classroom examples like Fibonacci where a closed form exists. It shows the explicit expression, explains how it was derived (characteristic equation or pattern recognition), and formats formulas in render-ready math markup so they scale responsively with the page’s math renderer. The tool also optionally displays intermediate steps: characteristic equation, roots, and constants solved from initial conditions.

How to Use this Explicit Formula Calculator

1. Enter the sequence definition: either an explicit pattern (e.g., \(a_n = 3n+2\)), an initial term plus rule (e.g., \(a_1=2, a_{n}=a_{n-1}+3\)), or a recurrence (e.g., \(a_n = 2a_{n-1}+3a_{n-2}\) with starting terms).
2. Choose the sequence type if known (arithmetic, geometric, linear recurrence), or leave it to auto-detect.
3. Click Calculate. The calculator returns the explicit formula \(a_n\), shows core formulas used in the derivation, and presents sample evaluations (e.g., \(a_{10}\)). For recurrences it solves the characteristic equation and fits constants to initial terms.

Formulas (render-ready)

Arithmetic sequence (first term \(a_1\), common difference \(d\)):
\( \; a_n = a_1 + (n-1)d \; \)

Geometric sequence (first term \(a_1\), ratio \(r\)):
\( \; a_n = a_1 \, r^{\,n-1} \; \)

Linear homogeneous recurrence with constant coefficients (roots \(r_i\)):
\( \; a_n = \sum_{i} C_i \, r_i^{\,n} \; \)

Example closed form for Fibonacci (Binet’s formula):
\( \; F_n = \dfrac{\varphi^{\,n} - \psi^{\,n}}{\sqrt{5}} \), where \( \varphi=\dfrac{1+\sqrt{5}}{2}, \; \psi=\dfrac{1-\sqrt{5}}{2} \).

Examples of Explicit Formula Calculator

Example 1 — Arithmetic: \(a_1=5, d=3\). Formula: \(a_n = 5 + (n-1)\cdot 3 = 3n+2\). So \(a_{7}=3\cdot7+2=23\).

Example 2 — Geometric: \(a_1=4, r=2\). Formula: \(a_n = 4\cdot 2^{\,n-1}\). So \(a_5 = 4\cdot2^{4}=64\).

Example 3 — Fibonacci: \(F_0=0, F_1=1\). Binet form gives \(F_n = \dfrac{\varphi^{\,n} - \psi^{\,n}}{\sqrt{5}}\), useful to compute large-index Fibonacci values in closed form.

Responsive formula notes

Formulas above use math markup that will scale and reflow when rendered by a math renderer on the page; place them in inline or display mode so they remain readable on mobile and desktop screens.

What is the difference between explicit and recursive formulas?

An explicit formula computes \(a_n\) directly as a function of n, while a recursive formula defines \(a_n\) in terms of previous terms.

Can every recursive sequence be converted to an explicit formula?

Not always. Many linear recurrences with constant coefficients have explicit solutions; more complex or non-linear recurrences may not have closed forms.

How do I find an explicit formula for an arithmetic sequence?

Use \(a_n = a_1 + (n-1)d\) where \(a_1\) is the first term and \(d\) is the common difference.

How do I find an explicit formula for a geometric sequence?

Use \(a_n = a_1 r^{\,n-1}\) where \(r\) is the common ratio and \(a_1\) the first term.

What is the characteristic equation method?

For linear recurrences, form a polynomial whose roots determine the general solution \(a_n=\sum C_i r_i^{n}\); constants are fixed by initial terms.

Is Binet’s formula exact for Fibonacci numbers?

Yes — Binet’s formula gives exact integer Fibonacci values despite involving irrational powers; rounding is not required if computed accurately.