Question 61 markMCQSection A

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 Arithmetic Progression

Options

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

Official answer

(a) √2 — successive terms differ by a constant √2 (e.g., 22−2=2=32−222\sqrt{2} - \sqrt{2} = \sqrt{2} = 3\sqrt{2} - 2\sqrt{2}).

Answer keywords:arithmetic progressioncommon differenceAP√2 seriesd = a2 - a1consecutive terms

Marking scheme

  1. 1.1 mark for correctly computing d=2d = \sqrt{2} and selecting option (a).

Hint

Common difference d = (any term) − (preceding term); compute 22−22\sqrt{2} - \sqrt{2}.

Quick oral answer

The common difference is √2, since each term is obtained by adding √2 to the previous term: 22−2=22\sqrt{2} - \sqrt{2} = \sqrt{2}.

Common mistakes

  1. 1.Subtracting terms in reverse order (a1−a2a_1 - a_2 instead of a2−a1a_2 - a_1), producing −2-\sqrt{2}.
  2. 2.Mistaking the coefficient sequence (1, 2, 3, 4) for the common difference and answering 1.
  3. 3.Forgetting to simplify 2\sqrt{2} terms correctly when subtracting surds.

Analysis and explanation

Tests finding the common difference of an AP with surd terms.


Concept

  • Common difference d = (any term) − (preceding term), and must be constant throughout the AP.

Key working

  • Given AP: 2,22,32,42\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, ...
  • d=22−2=2d = 2\sqrt{2} - \sqrt{2} = \sqrt{2}; check: d=32−22=2d = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2} — constant, confirming d=2d = \sqrt{2}.
  • The AP is √2 times the natural numbers (2×1,2×2,2×3\sqrt{2}\times1, \sqrt{2}\times2, \sqrt{2}\times3, ...), directly showing d=2d = \sqrt{2}.

Common mistakes

  • Picking the coefficient pattern (1,2,3,4) difference of 1 instead of the actual term difference.
  • Mistaking the second term (2√2) itself for the common difference.
  • Subtracting in reverse order (a1 − a2) and getting the wrong sign (−√2).

Real-world/exam link

  • Recognising AP structure in surd/scaled sequences helps quickly solve nth-term and sum-of-AP problems without recomputation.

Interesting facts

Arithmetic Progressions with irrational common differences, like 2\sqrt{2}, are commonly used in CBSE papers to test whether students truly understand the subtraction-based definition of 'd' rather than just pattern-spotting integers.

The sum formula for an AP, derived by Gauss as a schoolboy in the 18th century for the sequence 1 to 100, forms the basis for the SnS_n formulas taught in this chapter.

Question details

Q6 of 56 questions

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

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Frequently asked questions

How is the common difference of an AP calculated?

Subtract any term from the term immediately following it: d=a(n+1)−a(n)d = a(n+1) - a(n). The result must be the same for every consecutive pair for the sequence to be a valid AP.

Can the common difference of an AP be irrational, like 2\sqrt{2}?

Yes, the common difference can be any real number, including irrational numbers such as 2\sqrt{2}, as long as it remains constant throughout the sequence.