Arithmetic Progression (AP)

• The nth term of an AP: aₙ = a + (n−1)d.
• Sum of n terms: Sₙ = n/2 · [2a + (n−1)d] = n/2 · (first + last).
• Arithmetic mean and inserting AMs between two numbers.
• Property: if a, b, c are in AP then 2b = a + c.
• The difference-of-sums trick: aₙ = Sₙ − Sₙ₋₁.

Level 1 — AP definition and nth term

AP: A sequence where each term differs from the previous by a constant common difference d. General form: a, a+d, a+2d, … , a+(n−1)d.

nth term: aₙ = a + (n−1)d. Here a = first term, d = aₙ₊₁ − aₙ (constant).

Finding d: d = (last term − first term)/(n − 1) when n terms are known.

Sum formula: Sₙ = n/2 · [2a + (n−1)d]. Equivalent form: Sₙ = n/2 · (a + l) where l = last term = aₙ.

Arithmetic mean: AM of a and b is (a+b)/2. If m is inserted between a and b as AM, then a, m, b is AP with d = (b−a)/2.

Inserting n AMs between a and b: d = (b−a)/(n+1). The k-th AM = a + k·(b−a)/(n+1).

Level 2 — Advanced properties and problem-solving patterns

Property — three terms in AP: a, b, c in AP iff 2b = a + c. Use 3 terms as a−d, a, a+d (sum = 3a simplifies many problems).

Property — four terms in AP: Use a−3d, a−d, a+d, a+3d (sum = 4a, pairs symmetric about centre).

Difference-of-sums trick: aₙ = Sₙ − Sₙ₋₁ for n ≥ 2 (and a₁ = S₁). Vital when Sₙ is given as a quadratic in n — find d from coefficient of n².

If Sₙ = pn² + qn: Then aₙ = p(2n−1) + q (linear in n → confirms AP). First term a₁ = p + q, common difference d = 2p.

Sum of AP with last term l: Sₙ = n(a + l)/2. Number of terms n = (l − a)/d + 1.

Key JEE pattern: If terms of an AP are a₁, a₂, … then a₁ + aₙ = a₂ + aₙ₋₁ = … (equidistant terms have equal sums). Also useful: if Sₘ = Sₙ (m ≠ n), then S(m+n) = 0.

Reciprocal in HP — link: If a, b, c are in HP then 1/a, 1/b, 1/c are in AP.

NCERT spotlight

Standard result: Sum of first n natural numbers = n(n+1)/2 (AP with a=1, d=1). Sum of first n odd numbers = n². Tip: for Sₙ = pn² + qn, the sequence is always an AP — recognise by the quadratic form and use d = 2p.

The sum of first n terms of an AP is Sₙ = 3n² + 5n. Find the nth term and the common difference.

Step 1 — Use aₙ = Sₙ − Sₙ₋₁ for n ≥ 2:
         Sₙ = 3n² + 5n
         Sₙ₋₁ = 3(n−1)² + 5(n−1) = 3n² − 6n + 3 + 5n − 5 = 3n² − n − 2.
Step 2 — aₙ = Sₙ − Sₙ₋₁ = (3n² + 5n) − (3n² − n − 2) = 6n + 2.
Step 3 — Check a₁ = S₁ = 3(1)² + 5(1) = 8. Formula gives 6(1)+2 = 8. ✓
Step 4 — Common difference d = a₂ − a₁ = [6(2)+2] − [6(1)+2] = 14 − 8 = 6.
Step 5 — Alternatively: Sₙ = 3n² + 5n is quadratic in n with leading coefficient 3, so d = 2×3 = 6. ✓

How many terms of the AP 3, 7, 11, … must be taken so that the sum is 820?

Step 1 — Identify: a = 3, d = 4. Sₙ = n/2 [2(3) + (n−1)(4)] = n/2 [6 + 4n − 4] = n/2 [4n + 2] = n(2n+1).
Step 2 — Set Sₙ = 820: n(2n + 1) = 820 → 2n² + n − 820 = 0.
Step 3 — Discriminant: 1 + 4×2×820 = 1 + 6560 = 6561 = 81².
Step 4 — n = (−1 + 81)/(2×2) = 80/4 = 20. (Take positive root.)
Step 5 — Verify: S₂₀ = 20(40+1) = 20×41 = 820. ✓

1. Find the 20th term of the AP 5, 8, 11, …
2. Find the sum of first 15 terms of the AP 2, 5, 8, …
3. If a = 7, l = 63, Sₙ = 385, find n and d.
4. Insert 4 arithmetic means between 3 and 23.
5. Stretch: Prove that if Sₘ = Sₙ for an AP (m ≠ n), then S(m+n) = 0, and find the term that is zero.

Open the Practice tab for graded questions on Sequences & Series — AP.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
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• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
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• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

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Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.