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Recursive Rule

A recursive rule defines each term of a sequence using one or more previous terms, along with a starting term.

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Student-friendly explanation

A recursive rule does not usually give a far-away term directly. It tells how to move from one term to the next. For example, if a1 = 4 and an = an-1 + 3, the sequence is 4, 7, 10, 13, and so on.

How to write this in exams

  1. 1

    Start with the exact idea

    A recursive rule defines each term of a sequence using one or more previous terms, along with a starting term.

  2. 2

    Then show how to use it

    Write the starting term. Apply the given operation to get the next term. Repeat carefully. Stop at the required position. Do not skip intermediate terms unless a direct formula is also known.

  3. 3

    Add one concrete example

    If a1 = 2 and each next term is double the previous term, then the sequence is 2, 4, 8, 16.

  4. 4

    Avoid this incomplete answer

    For a1 = 6 and an = an-1 + 5, some students write a2 = 5, but 5 is the difference, not the second term.

Definition

A recursive rule defines each term of a sequence using one or more previous terms, along with a starting term.

Example

If a1 = 2 and each next term is double the previous term, then the sequence is 2, 4, 8, 16.

Rule to remember

Typical recursive form: a1 = starting value, an = an-1 + d for n greater than 1.

Memory hook

Recursive rule says: use the previous term, then move one step.

Examples and method

Worked example

Given a1 = 6 and an = an-1 + 5, the terms are a1 = 6, a2 = 11, a3 = 16, and a4 = 21.

Method to apply

Write the starting term. Apply the given operation to get the next term. Repeat carefully. Stop at the required position. Do not skip intermediate terms unless a direct formula is also known.

Diagram support

An arrow diagram from one term to the next can show how the same operation is repeated.

How CBSE asks it

Students may be asked to generate the first few terms from a recursive rule or write a recursive rule from a given pattern.

Avoid common mistakes

Common confusion

Students write only the changing rule but forget the starting term, so the sequence cannot be fixed uniquely.

Common wrong answer

For a1 = 6 and an = an-1 + 5, some students write a2 = 5, but 5 is the difference, not the second term.

Exam tip

In recursive questions, mention both the first term and the rule for getting the next term.

Quick check

Why is the starting term necessary in a recursive rule?

The starting term is necessary because the rule tells how to move ahead, but the first value decides where the sequence begins.

Study the recursive rule diagram carefully

Use the labelled diagram to keep recursive rule clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of recursive rule in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on recursive rule.

Revision cue

Revise recursive rule through the labels before writing the answer.

Answer writing and exam use

1-mark answer

A recursive rule defines each term of a sequence using one or more previous terms, along with a starting term.

2-mark answer

A recursive rule defines each term of a sequence using one or more previous terms, along with a starting term. Typical recursive form: a1 = starting value, an = an-1 + d for n greater than 1. If a1 = 2 and each next term is double the previous term, then the sequence is 2, 4, 8, 16.

3-mark answer

A recursive rule does not usually give a far-away term directly. It tells how to move from one term to the next. For example, if a1 = 4 and an = an-1 + 3, the sequence is 4, 7, 10, 13, and so on. Typical recursive form: a1 = starting value, an = an-1 + d for n greater than 1. Given a1 = 6 and an = an-1 + 5, the terms are a1 = 6, a2 = 11, a3 = 16, and a4 = 21. Students may be asked to generate the first few terms from a recursive rule or write a recursive rule from a given pattern. For a1 = 6 and an = an-1 + 5, some students write a2 = 5, but 5 is the difference, not the second term.
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