medium difficulty5 min

Recursive Rule

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

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

Understand it

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.

Example

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

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

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

Method you can reuse

1Write the starting term.
2Apply the given operation to get the next term.
3Repeat carefully.
4Stop at the required position.
5Do not skip intermediate terms unless a direct formula is also known.

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is the starting term necessary in a recursive rule?

Think first, then show answer

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

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.

Don't mix this up

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

Exam tip

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

Recursive Rule diagram

Open full screen to read every label clearly.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

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.

1
Enough for 1 mark
A recursive rule defines each term of a sequence using one or more previous terms, along with a starting term.
2
Add for 2 marks
Example: If a1 = 2 and each next term is double the previous term, then the sequence is 2, 4, 8, 16.
3
Add for 3 marks
A recursive rule does not usually give a far-away term directly.

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