Question 61 markMCQSection A

  1. The common difference of the AP : 2,22,32,42\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, ..... is : (a) √2 (b) 1 (c) 2√2 (d) -√2

Arithmetic Progressions
Common difference of an AP

Options

(A)2\sqrt{2}
(B)1
(C)222\sqrt{2}
(D)−2-\sqrt{2}

Official answer

(a) 2\sqrt{2} — d=a2−a1=22−2=2d = a_2 - a_1 = 2\sqrt{2} - \sqrt{2} = \sqrt{2}, confirmed constant across all consecutive terms.

Answer keywords:common differencearithmetic progressiond = a2 - a12√2 - √2√2multiples of √2constant difference

Marking scheme

  1. 1.1 mark: correct option (a) 2\sqrt{2}.
  2. 2.Expected reasoning: d=a2−a1=22−2=2d = a_2 - a_1 = 2\sqrt{2} - \sqrt{2} = \sqrt{2}, verified as constant across the sequence.

Hint

Subtract the first term from the second: 22−22\sqrt{2} - \sqrt{2}.

Quick oral answer

The common difference is the second term minus the first: 22−2=22\sqrt{2} - \sqrt{2} = \sqrt{2}, and it stays constant throughout the sequence.

Common mistakes

  1. 1.Reading off the second term 222\sqrt{2} as the common difference instead of subtracting consecutive terms.
  2. 2.Ignoring the 2\sqrt{2} factor and using the coefficients 1, 2, 3, 4 to get a difference of 1.
  3. 3.Assigning a negative sign (−2-\sqrt{2}) even though the terms are increasing, so the difference is positive.

Analysis and explanation

This MCQ checks the method for finding the common difference of an AP with surd terms.


Concept

  • Common difference d=an+1−and = a_{n+1} - a_n (any term minus its preceding term).
  • For 2,22,32,42,…,d=22−2=2\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, \ldots, d = 2\sqrt{2} - \sqrt{2} = \sqrt{2}.

Key points

  • Verified consistent: 32−22=23\sqrt{2} - 2\sqrt{2} = \sqrt{2} and 42−32=24\sqrt{2} - 3\sqrt{2} = \sqrt{2}.
  • Terms are 2×1,2×2,2×3\sqrt{2} \times 1, \sqrt{2} \times 2, \sqrt{2} \times 3, ... so √2 is the natural step size.

Common mistakes

  • Reading off the coefficients 1,2,3,4 as the difference (giving 1) and ignoring the common factor √2.
  • Picking 222\sqrt{2} (the second term, not the difference) or the wrong sign −2-\sqrt{2} for an increasing sequence.

Interesting facts

An AP can have an irrational common difference like 2\sqrt{2}; the definition only requires the difference between consecutive terms to be CONSTANT, not that it be a whole number.

This AP is just 2\sqrt{2} times the natural-number sequence 1, 2, 3, 4, ..., so its nth term is an=n2a_n = n\sqrt{2} — scaling a simple AP by a constant keeps it an AP.

The concept of arithmetic progressions dates back over a thousand years; the young Carl Friedrich Gauss famously summed the AP 1+2+…+100=50501+2+\ldots+100 = 5050 in seconds as a schoolboy.

Question details

Q6 of 53 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Basic)
Year
2026
Section
Section A
Marks
1
Type
MCQ
Chapter
Arithmetic Progressions
Topic
Common difference of an AP
Bloom's level
apply
Syllabus topic
Unit 2: Algebra — Arithmetic Progressions: Common Difference of an AP
Author
CBSE
Reviewed by
Padhantu Academic Team

Spotted a mistake or something unclear?

Tell us and we will fix it.

Frequently asked questions

How do you find the common difference of an AP?

Subtract any term from the term that follows it: d=an+1−and = a_{n+1} - a_n. For 2,22,32,…,d=22−2=2\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, \ldots, d = 2\sqrt{2} - \sqrt{2} = \sqrt{2}, and you can verify it is the same for every consecutive pair.

Can the common difference of an AP be irrational?

Yes. The only requirement for an AP is that the difference between consecutive terms is constant. That constant can be irrational, like 2\sqrt{2}, or negative, or a fraction — it just has to stay the same throughout.

Why isn't the common difference 222\sqrt{2} here?

222\sqrt{2} is the second term of the sequence, not the gap between consecutive terms. The common difference is 22−2=22\sqrt{2} - \sqrt{2} = \sqrt{2}. Reading a term as the difference is a common trap.