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MathsMediumClass 11

Geometric Progression + Infinite GP

Sequences & Series

46

JEE Qs

10%

Hard

80

min

Master the condition for convergence of infinite GP and strategically choose terms (e.g., a/r, a, ar) to simplify complex algebraic problems.

🧮 Key Formulas

General term of a GP: a_n = a * r^(n-1)
Sum of n terms of a GP (if r ≠ 1): S_n = a * (r^n - 1) / (r - 1) OR S_n = a * (1 - r^n) / (1 - r)
Sum of infinite GP (if |r| < 1): S_infinity = a / (1 - r)
Geometric Mean of n positive numbers (a_1, a_2, ..., a_n): G = (a_1 * a_2 * ... * a_n)^(1/n)

✅ Key Points for JEE

  • 1Always check the common ratio 'r' and the first term 'a' to correctly identify and apply GP formulas.
  • 2The condition |r| < 1 is crucial for an infinite GP to converge to a finite sum; otherwise, the sum diverges.
  • 3When three terms are in GP, it is often useful to assume them as a/r, a, ar to simplify products; for four terms, a/r^3, a/r, ar, ar^3 can be useful.
  • 4Problems often involve combining GP with other series types (AP/HP) or algebraic conditions (equations, inequalities) to find unknowns.
  • 5Be mindful of the cases r=1 (series is a, a, a... sum is na) and r=-1 (series is a, -a, a, -a... sum oscillates).

⚠️ Common Mistakes

  • ✕Forgetting or misapplying the condition |r| < 1 for the sum of an infinite GP, leading to incorrect divergence/convergence assumptions.
  • ✕Incorrectly calculating 'n' (number of terms) or 'r' (common ratio), especially when solving logarithmic equations or algebraic manipulations.
  • ✕Using the sum of n terms formula (S_n) when r=1; in this case, S_n = na.

📝 Practice Questions

See all

Q12.The remainder, when 7103 is divided by 23 , is equal to : (1) 6 (2) 17 (3) 9 (4) 14

2025·MCQMedium

Q22.Let a1, a2, … , a2024 be an Arithmetic Progression such that a1 + (a5 + a10 + a15 + … + a2020) + a2024 = 2233. Then a1 + a2 + a3 + … + a2024 is equal to _______ 1 2 3 , then α is equal to ________ (3x + t = 5eα ( 85 )

2025·NumericalMedium

Q1. Let a1, a2, a3, … be a G.P. of increasing positive terms. If a1a5 = 28 and a2 + a4 = 29, then a6 is equal to: (1) 628 (2) 812 (3) 526 (4) 784 = 0. If x(1) = 1, then x ( 12 ) is :

2025·MCQMedium

Q15.If ∑nr=1 Tr = (2n−1)(2n+1)(2n+3)(2n+5)64 , then limn→∞∑nr=1 ( Tr1 ) (1) 0 (2) 23 (3) 1 (4) 13

2025·MCQHard

Q13.Suppose that the number of terms in an A.P. is 2k, k ∈N . If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then k is equal to : (1) 6 (2) 5 (3) 8 (4) 4 y+2

2025·MCQMedium

Q1. If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to (1) −1080 (2) −1020 (3) −1200 (4) −120

2025·MCQEasy

NCERT Chapters

  • Class 11 Maths Ch 9: Sequences and Series