medium difficulty5 min

AP from linear patterns

A linear pattern can be turned into an AP when the terms increase or decrease by equal steps in a real-life or number pattern.

In one line: Equal step pattern means AP, not growth by a fixed factor.

Practice This Concept

Understand it

Many AP questions come from patterns in tiles, rows, seats, money saved, or dots. If the pattern changes by the same fixed amount each time, it forms an AP. Students should learn to convert the situation into terms and then identify the first term and common difference.

Example

If a student saves 10, 20, 30, 40 rupees each week, the savings form an AP with first term 10 and common difference 10.

Formula and rule

Translate the pattern into terms a, a + d, a + 2d, ... before using AP formulas.

See it worked out

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

Worked example

A pattern of 2 chairs in the first row, 5 in the second, 8 in the third, and 11 in the fourth row forms an AP with a = 2 and d = 3.

Method you can reuse

1Read the pattern carefully.
2Check the change between consecutive terms.
3Write the first term and d.
4Apply the AP formula if needed.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

When does a real-life pattern become an arithmetic progression?

Think first, then show answer

A real-life pattern becomes an arithmetic progression when each new term changes by the same fixed amount. Equal additions or equal subtractions turn the pattern into an AP.

Common wrong answer

A common wrong answer is to treat the pattern as a multiplication table or geometric pattern instead of an AP.

Don't mix this up

Students may write the pattern as multiplication when the actual growth is by equal addition.

Exam tip

Look for a repeated increase or decrease by the same amount before applying AP formulas.

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

CBSE may give a seating, saving, or arranging pattern and ask students to identify it as an AP or find a later term.

1
Enough for 1 mark
A linear pattern can be turned into an AP when the terms increase or decrease by equal steps in a real-life or number pattern.
2
Add for 2 marks
If the pattern changes by the same fixed amount each time, it forms an AP.
3
Add for 3 marks
A pattern of 2 chairs in the first row, 5 in the second, 8 in the third, and 11 in the fourth row forms an AP with a = 2 and d = 3.

MCQ Quiz

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