Calculate recursive sequence terms from coefficient r, constant c, and initial value a(0), including generated lists, nth term, and partial sum.

Rule: a(n) = r × a(n−1) + c, starting at a(0).

Up to 200 displayed terms; the first is a(0).

Related Calculators

Recursive Rule Formula

The calculator uses a(n) = r × a(n−1) + c with initial term a(0). The following formula is its arithmetic special case, with r = 1 and c = d. Variables:

a(n) = a(n - 1) + d
  • a(n) is the nth term in the sequence
  • a(n-1) is the previous term in the sequence
  • d is the common difference between the terms

For this arithmetic special case, take the previous term in the sequence and add the common difference. This will give you the next term in the sequence.

What is a Recursive Rule?

A recursive rule is a mathematical rule or formula that uses the output of one step to determine the output of the next step. It is a way of defining a sequence or a pattern of numbers where each term is a function of the preceding term. This rule is often used in computer programming and mathematics, particularly in algorithms and functions, to solve complex problems by breaking them down into simpler, repetitive steps.

How to Calculate Recursive Rule?

The following steps illustrate the arithmetic special case: a(n) = a(n-1) + d. In the calculator, use coefficient r = 1, additive constant c = d, and the initial term a(0).

  1. First, determine the value of the previous term, a(n-1).
  2. Next, determine the value of the common difference, d.
  3. Next, substitute the values of a(n-1) and d into the formula: a(n) = a(n-1) + d.
  4. Finally, calculate the value of the nth term, a(n).
  5. After calculating the result, check your answer with the given sequence or pattern.

Example Problem:

Use the following variables as an example problem to test your knowledge.

Previous term, a(n-1) = 5

Common difference, d = 3