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MathsMediumClass 11

GP — nth term, sum, infinite GP, GM

Sequences & Series

14

JEE Qs

8%

Hard

60

min

Master the identification of 'a' and 'r', pay close attention to the `|r| < 1` condition for infinite GPs, and practice strategic term assumption for product-based problems.

🧮 Key Formulas

a_n = a * r^(n-1)
S_n = a * (r^n - 1) / (r - 1) (for r != 1)
S_n = n * a (for r = 1)
S_infinity = a / (1 - r) (for |r| < 1)
GM(x, y) = sqrt(x * y)
GM(x_1, x_2, ..., x_n) = (x_1 * x_2 * ... * x_n)^(1/n)

✅ Key Points for JEE

  • 1Always correctly identify the first term 'a' and the common ratio 'r' from the problem statement, as these are fundamental to all GP calculations.
  • 2The condition `|r| < 1` is absolutely essential for the sum of an infinite geometric progression to converge; otherwise, the sum diverges.
  • 3When a product of terms in a GP is given, assume the terms strategically (e.g., for 3 terms: `a/r, a, ar`; for 4 terms: `a/r^3, a/r, ar, ar^3`) to simplify calculations significantly.
  • 4Geometric Mean is defined for positive numbers. Remember the AM-GM inequality: `GM <= AM` for positive numbers, with equality only if all numbers are equal.
  • 5A key property of GP is that the square of any term is the product of its preceding and succeeding terms: `t_k^2 = t_(k-1) * t_(k+1)`.

⚠️ Common Mistakes

  • ✕Forgetting or misapplying the `|r| < 1` condition for the convergence of an infinite GP, leading to incorrect sums or conclusions about divergence.
  • ✕Algebraic errors, especially with negative common ratios or when dealing with fractional powers in geometric mean calculations.
  • ✕Confusing the formulas or conditions for Geometric Progression with those of Arithmetic Progression or other series types.
  • ✕Incorrectly assuming geometric mean can be applied to negative numbers or not understanding its definition for multiple numbers.

📝 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