Geometric Progression (GP)

• The nth term of a GP: aₙ = a·r^(n−1).
• Sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1.
• Infinite GP: S∞ = a/(1 − r) for |r| < 1.
• Geometric mean and the AM ≥ GM ≥ HM inequality chain with proof.

Level 1 — GP definition and formulas

GP: A sequence where each term is r times the previous. General form: a, ar, ar², ar³, … Common ratio r = aₙ₊₁/aₙ (constant).

nth term: aₙ = a·r^(n−1).

Sum of n terms:

• r ≠ 1: Sₙ = a(rⁿ − 1)/(r − 1) = a(1 − rⁿ)/(1 − r) (both forms equivalent).
• r = 1: Sₙ = na (all terms equal).

Infinite GP: If |r| < 1, as n → ∞, rⁿ → 0, so S∞ = a/(1 − r). If |r| ≥ 1, the infinite sum diverges.

Geometric mean: GM of a and b is √(ab). If G is inserted between a and b as GM, then a, G, b is GP with r = √(b/a).

Inserting n GMs between a and b: r = (b/a)^(1/(n+1)). The k-th GM = a·r^k.

Level 2 — Properties and AM ≥ GM ≥ HM

Three terms in GP: Use a/r, a, ar (product = a³ → product is a cube, simplifies problems).

Four terms in GP: Use a/r³, a/r, ar, ar³ (product = a⁴).

Product property: If a₁, a₂, …, a₂ₙ₋₁ are in GP, then a₁·a₂ₙ₋₁ = a₂·a₂ₙ₋₂ = … = aₙ² (equidistant terms from centre have equal products).

AM ≥ GM inequality: For positive reals a, b: (a+b)/2 ≥ √(ab). Equality iff a = b. Proof: (√a − √b)² ≥ 0 → a − 2√(ab) + b ≥ 0 → (a+b)/2 ≥ √(ab). □

GM ≥ HM inequality: For positive reals a, b: √(ab) ≥ 2ab/(a+b). Proof: AM ≥ GM applied to 1/a and 1/b gives (1/a + 1/b)/2 ≥ √(1/(ab)) → 2/(HM) ≥ 1/GM → GM ≥ HM.

Full chain AM ≥ GM ≥ HM: (a+b)/2 ≥ √(ab) ≥ 2ab/(a+b), equality throughout iff a = b.

JEE optimisation pattern: To minimise a + b given ab = constant k, use AM ≥ GM: a + b ≥ 2√(ab) = 2√k, minimum at a = b = √k.

Sum of infinite recurring decimals: 0.333… = 3/10 + 3/100 + … = (3/10)/(1 − 1/10) = 1/3. (Infinite GP.)

NCERT spotlight

Identify GP vs AP: check if ratios are constant (GP) or differences are constant (AP). For S∞, check |r| < 1 first — if forgotten, you'll sum a divergent series. GM of a set of n numbers: (a₁·a₂·…·aₙ)^(1/n).

Find the sum of the infinite GP: 1 + 1/3 + 1/9 + 1/27 + …

Step 1 — Identify: a = 1, r = (1/3)/1 = 1/3. Check |r| = 1/3 < 1. ✓
Step 2 — Apply S∞ = a/(1 − r) = 1/(1 − 1/3) = 1/(2/3) = 3/2.
Step 3 — Verify partial sums converge: S₁=1, S₂=4/3≈1.33, S₃=13/9≈1.44 → approaching 3/2=1.5. ✓

Three numbers in GP have product 216 and their sum is 19. Find the numbers.

Step 1 — Let the three numbers be a/r, a, ar.
Step 2 — Product: (a/r)·a·(ar) = a³ = 216 → a = 6.
Step 3 — Sum: 6/r + 6 + 6r = 19 → 6/r + 6r = 13.
Step 4 — Multiply by r: 6r² − 13r + 6 = 0.
Step 5 — Discriminant: 169 − 144 = 25. r = (13 ± 5)/12 → r = 3/2 or r = 2/3.
Step 6 — For r = 3/2: terms are 4, 6, 9. For r = 2/3: terms are 9, 6, 4 (same set, reversed).
Step 7 — Answer: the three numbers are 4, 6, 9.

1. Find the 8th term of GP: 2, 6, 18, …
2. Find the sum of 6 terms of GP: 1, −2, 4, −8, …
3. Find S∞ for GP: 8, 4, 2, 1, …
4. If AM = 10 and GM = 8 for two positive numbers, find the numbers.
5. Stretch: Prove that if a, b, c, d are in GP then (a² + b² + c²)(b² + c² + d²) = (ab + bc + cd)². (Use GP property aₙ₋₁·aₙ₊₁ = aₙ².)

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

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• 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."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• 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.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

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.