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MathsMediumClass 11

AGP — Arithmetico-Geometric Progression

Sequences & Series

14

JEE Qs

8%

Hard

60

min

Master the 'S - RS' technique for AGP summation, as understanding the method is more crucial and flexible than memorizing complex sum formulas for 'n' terms.

🧮 Key Formulas

General term a_n = [A + (n-1)D] * R^(n-1)
Sum of n terms S_n = (A / (1-R)) + (DR * (1-R^(n-1)) / (1-R)^2) - ((A + (n-1)D) * R^n / (1-R)) (for R ≠ 1)
Sum to infinity S_infinity = A/(1-R) + DR/(1-R)^2 (for |R| < 1)

✅ Key Points for JEE

  • 1An Arithmetico-Geometric Progression (AGP) is a series where each term is the product of the corresponding term of an Arithmetic Progression (AP) and a Geometric Progression (GP).
  • 2The sum of 'n' terms (S_n) of an AGP is typically found using the 'S - RS' method: write the sum S_n, multiply it by the common ratio 'R' of the GP, shift the terms by one position, and then subtract the two series.
  • 3After applying the 'S - RS' method, the resultant series typically simplifies into a combination of a single term and a standard Geometric Progression, which can then be easily summed.
  • 4The sum to infinity (S_infinity) of an AGP exists if and only if the absolute value of the common ratio 'R' of the GP component is less than 1 (|R| < 1).

⚠️ Common Mistakes

  • ✕Making algebraic errors, especially with signs and exponents, while performing the S - RS subtraction and subsequent simplification.
  • ✕Incorrectly identifying the common ratio 'R' of the GP or the common difference 'D' of the AP component of the AGP.
  • ✕Forgetting or misapplying the condition |R| < 1 when calculating the sum to infinity, leading to an incorrect assumption of convergence.
  • ✕Errors in summing the resulting GP after the S - RS step, particularly when determining the number of terms in the new GP.

📝 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