Sequences Series

Sequences Series

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A sequence is an ordered list of numbers whose n-th term follows a definite rule, written a_n. A series is the sum Σa_n of those terms. Two families dominate CUET UG: the Arithmetic Progression (AP) with constant difference d = a_n − a_{n−1}, and the Geometric Progression (GP) with constant ratio r = a_n / a_{n−1}.

Must-know formulas:

• AP n-th term: a_n = a + (n − 1)d; sum: S_n = n/2 [2a + (n − 1)d]
• GP n-th term: a_n = ar^(n−1); sum: S_n = a(1 − r^n)/(1 − r) for r ≠ 1
• Infinite GP: S_∞ = a/(1 − r) only when |r| < 1
• AM = (a+b)/2, GM = √(ab), and for positives AM ≥ GM ≥ HM

Quick exam cues: 1+1 = 2 is the smallest AP; 2, 4, 8, … is a GP with r = 2; Σk = n(n+1)/2.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Definitions and Notation

An AP is a sequence where every consecutive pair differs by the same number d: a, a+d, a+2d, … Its general term is a_n = a + (n−1)d, where a is the first term. A GP instead multiplies by a fixed ratio r: a, ar, ar², … giving a_n = ar^(n−1). The series sum S_n of an AP equals n/2 [2a + (n−1)d] and can also be written as n/2 (a + l) where l is the last term. For a GP, S_n = a(1 − r^n)/(1 − r) when r ≠ 1; if r = 1, S_n collapses to na (a common trap on CUET).

Why S_n of an AP is Quadratic

Because a_n itself is linear in n, summing from 1 to n yields a quadratic. Concretely, a_n = S_n − S_{n−1}, so if S_n = An² + Bn + C, then a_n = 2An + (B − A), confirming the linear n-th term.

Means and Insertion

Given two numbers a and b, the Arithmetic Mean (AM) is (a+b)/2 and the Geometric Mean (GM) is √(ab). To insert n AMs between a and b, treat the full list as an AP of length n+2 with difference d = (b−a)/(n+1). For n GMs, use ratio r = (b/a)^(1/(n+1)). The classical inequality AM ≥ GM ≥ HM holds only for strictly positive numbers — examiners often test this positivity condition.

Special Natural-Number Sums

These identities are asked almost every year:

Sum	Closed form
1 + 2 + … + n	n(n+1)/2
1² + 2² + … + n²	n(n+1)(2n+1)/6
1³ + 2³ + … + n³	[n(n+1)/2]²

CUET Question Patterns

Expect 2–3 MCQs: computing a_n or S_n given a and d (or a and r); spotting whether a given list is AP or GP; one question on infinite GP convergence (test |r| < 1); an AM-GM inequality problem assuming positive terms. Numerical traps include off-by-one errors where the “first term” is actually a_2, and applying the GP sum formula at r = 1.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Infinite Series Convergence

The expression S_∞ = a/(1−r) is meaningful only when |r| < 1; otherwise partial sums diverge to ±∞ or oscillate. This is why a question such as “sum of 1 + 2 + 4 + 8 + …” has no finite answer, while 1 + 1/2 + 1/4 + 1/8 + … = 2. Memorise the boundary: at r = 1 you get the harmonic-divergent case na; at r = −1 you get the alternating series 1, −1, 1, −1, … whose partial sums oscillate and never settle.

Arithmetico-Geometric Progression (AGP)

A sequence whose n-th term has the form (a + (n−1)d)·r^(n−1) is an AGP. The sum to n terms is obtained by the difference method: multiply S_n by r, subtract, and use a telescoping cancellation. AGPs appear occasionally in CUET mixed-concepts papers, especially when an AP multiplier meets a GP factor.

Edge Cases and Common Mistakes

• Index confusion. The term called “the 3rd term” in a question may actually be a_2 if the stated list starts at index 0. Always anchor a_n = a + (n−1)d with the right a.
• AM-GM on negatives. The inequality fails or reverses for non-positive numbers; e.g., a = −4, b = −9 gives AM = −6.5, GM = √36 = 6, so GM > AM. Always check the sign.
• Sum vs n-th term. S_n is cumulative; a_n is incremental. Using the S_n formula when the question asks for a_n (or vice versa) is the single most common error.
• Insertion formulas. d and r for insertion problems depend on n+1, not n — a frequent off-by-one.

Adjacent Topics

Sequences & Series connects to Binomial Theorem (coefficients form APs in particular expansions), Mathematical Induction (proving the natural-number sums), and Limits (S_∞ of a GP is a limit). Mastering the AP–GP dichotomy here makes coordinate geometry and probability’s geometric-distribution questions easier.

Practice Prompts

1. If the 7th and 13th terms of an AP are 10 and 16, find S_20. (Hint: solve for a and d first; expect S_20 = 590.)
2. Find the sum to infinity of the GP: 0.15, 0.0015, 0.000015, … (a = 0.15, r = 0.01; S_∞ = 0.15/0.99 = 5/33.)

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