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Principle deep dive

AP three-term: 2b = a + c

If a, b, c are in AP, then 2b = a + c. The bank disguises this across Lines (collinearity, family of lines), Logarithms, M&D, P&C, Probability, Properties of Triangle, Inverse Trig, and Trig Identities — nine chapters of cross-chapter reach.

questions in the bank
32
tagged HARD
28%
chapter spread
9
worked examples below
4

When to reach for it

Three terms are in AP — or you can rewrite them to be in AP — and you can use 2b = a + c to close the system.

Why this principle matters

The AP three-term identity is the simplest unifying relation in the bank: if a, b, c are in AP, then 2b = a + c. That's all. But the principle's leverage comes from how NDA disguises the three terms.

Sometimes a, b, c are explicit numbers in an AP question. Sometimes they're logarithms (log p, log q, log r in AP). Sometimes they're binomial coefficients (C(n, 4), C(n, 5), C(n, 6) in AP). Sometimes they're sides of a triangle. Whatever the surface, the relation is the same — and combined with one other equation, you can solve.

Don't conflate this with the AP sum formulas (Sₙ = n/2 · [2a + (n−1)d]) — those are a separate skill. The three-term identity is the algebraic core; the sum formulas are computational. NDA tests both, often in the same question.

4 worked examples from the bank

Each example demonstrates the principle on a real past-year question. Click to reveal the answer, then the solution.

Example 1Sequence & SeriesEASY
pp, qq, rr and ss are in AP such that p+s=8p+s=8 and qr=15qr=15. What is the difference between largest and smallest numbers?

[Q26 · Apr · 2023]

Example 2Sequence & SeriesEASY
If x2,x,−8x^2, x, -8 are in AP, then which one of the following is correct?

[Q52 · Apr · 2021]

Example 3Properties of TriangleMODERATE
If the angles of a triangle ABCABC are in AP and b:c=3:2b : c = \sqrt{3} : \sqrt{2}, then what is the measure of angle AA ?

[Q62 · Sep · 2019]

Example 4Permutation & CombinationHARD
If C(n,4)C(n,4), C(n,5)C(n,5) and C(n,6)C(n,6) are in AP, then what is the value of nn?

[Q17 · Sep · 2021]

Variants to recognise

Same principle, different surfaces. Pattern-match these on test day.

  • 2b = a + c (basic)

    For any three terms a, b, c in AP. The single equation that closes most AP questions.

  • Sₙ = n/2 · [2a + (n − 1)d]

    Sum of first n terms. Pairs with the three-term identity in compound questions.

  • log a, log b, log c in AP ⇔ b² = ac (i.e. a, b, c in GP)

    The log-AP-to-GP bridge. Recognise this when you see log appear inside an AP statement.

  • AP within an AP

    If the kᵗʰ terms of two APs are in AP, the original APs share a relationship. Tested in P&C and Logs.

Drill every ap three-term: 2b = a + c question

32 questions from the bank — paginated, with cart and Word-export support.

Drill the 32 questions

Related principles

Often combined with this one — drill these next if you found the examples above tractable.