1 markMCQSection A

The common difference of the AP : $\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}$ , ..... is :

(a) √2 (b) 1 (c) 2√2 (d) −√2

Arithmetic Progressions

Common Difference of an Arithmetic Progression

Options

(A)

\sqrt{2}

(B)1

(C)

2\sqrt{2}

(D)

-\sqrt{2}

Official Answer

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

arithmetic progressioncommon differenceAP√2 seriesd = a2 - a1consecutive terms

Marking Scheme

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

Hint

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

Quick Oral Answer

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

Analysis & 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: $\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}$ , ...
$d = 2\sqrt{2} - \sqrt{2} = \sqrt{2}$ ; check: $d = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2}$ — constant, confirming $d = \sqrt{2}$ .
The AP is √2 times the natural numbers ( $\sqrt{2}\times1, \sqrt{2}\times2, \sqrt{2}\times3$ , ...), directly showing $d = \sqrt{2}$ .

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.

Common Mistakes

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

Interesting Facts

Arithmetic Progressions with irrational common differences, like $\sqrt{2}$ , 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 $S_n$ formulas taught in this chapter.

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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: $d = a(n+1) - a(n)$ . 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 $\sqrt{2}$ ?

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