What are Sequences, Series and Progressions?

Sequences, Series, and Progressions form the core of this topic. Understanding the distinction between these three is the first step to mastering the chapter.

Term Definition Example
Sequence An ordered list of numbers arranged according to a definite rule. Can be finite or infinite. {2, 4, 6, 8, 10, ...}
Series The sum of the terms of a sequence. Adding up the elements gives a series. 1 + 2 + 3 + ... + n = n(n+1)/2 
Progression  A special sequence where each term is obtained from the previous one by a fixed operation - addition (AP), multiplication (GP), or reciprocal (HP).  2, 5, 8, 11 (AP with d = 3)

Progressions are one of the important topics for CAT. To ensure you are not missing any similarly important topics, checking the CAT Exam Syllabus is advised.

CAT Progressions and Series Topic Weightage Over Past 8 Years

The table below gives the year-wise number of questions from Progressions and Series in CAT QA across all slots. This data confirms that the topic carries consistent and significant weight every year.

Year Weightage (No. of Questions)  Difficulty Level  Key Focus Area
CAT 2025  5 Medium–Hard AP sums, Infinite GP, Odd number series
CAT 2024 5 Medium AP terms, Recurrence sequences, Floor function
CAT 2023 7 Medium–Hard AP with logarithms, Mixed AP+GP, Common terms
CAT 2022 5 Medium AP averages, nth term, Particle growth
CAT 2021 5 Medium AP integers, Recurrence, Alternating series
CAT 2020 2 Easy–Medium GP common ratio, Recursive sequences
CAT 2019 6 Medium AP rationalisation, Alternating series, Population GP 
CAT 2018 5 Medium Series sums, AM averages, AP products, GP ratio
Student Tip: CAT 2023 was the heaviest year for this topic with 7 questions. Never deprioritise Progressions - even in a relatively easy paper, 5 questions from a single chapter can change your QA score significantly.

Also Check | CAT Mock Test series

CAT Progressions and Series Formulas PDF

CAT Progressions and Series is one of the most formula-heavy topics in Quantitative Aptitude. A clear understanding of each formula - and knowing when to apply it - is what separates a 90+ scorer from an average performer on this topic. Click on the link below to download the complete Progressions and Series Formulas PDF.

1. Arithmetic Progression (AP) - Formulas and Properties

If the difference between any two consecutive terms is constant, the terms are said to be in AP. This constant is called the common difference (d).

General form: a, a+d, a+2d, a+3d, ...

AP Formulas
nth Term T(n) = a + (n-1)d
Sum of n Terms S(n) = (n/2) × [2a + (n-1)d]
Sum (first & last term) S(n) = (n/2) × [First Term + Last Term]
Number of Terms n = [(Last Term - First Term) / d] + 1
Arithmetic Mean If a, b, c are in AP → b = (a + c) / 2

Properties of Arithmetic Progression: If a, b, c, d, ... are in AP and k is a constant, then:

a–k, b–k, c–k, ... are also in AP
ak, bk, ck, ... are also in AP (k ≠ 0)
a/k, b/k, c/k, ... are also in AP (k ≠ 0)
If S(n) = kn², the sequence is an AP

2. Geometric Progression (GP) - Formulas and Properties

If in a succession of numbers the ratio of any term to the previous term is constant, the numbers are said to be in GP. This constant is called the common ratio (r).

General form: a, ar, ar², ar³, ...

GP Formulas
nth Term T(n) = a × r^(n-1)
Sum of n Terms (r > 1) S(n) = a(r^n - 1) / (r - 1)
Sum of n Terms (r < 1) S(n) = a(1 - r^n) / (1 - r)
Infinite GP Sum S∞ = a / (1 - r) [valid only when |r| < 1]
Geometric Mean If a, b, c are in GP → b² = ac

Properties of Geometric Progression: If a, b, c, d, ... are in GP and k is a constant, then:

ak, bk, ck, ... are also in GP
a/k, b/k, c/k, ... are also in GP
log a, log b, log c, ... are in AP
1/a, 1/b, 1/c, ... are also in GP

3. Harmonic Progression (HP) - Formulas and Properties

A sequence is in Harmonic Progression if the reciprocals of its terms form an Arithmetic Progression. If a, b, c are in HP, then 1/a, 1/b, 1/c are in AP.

HP Formulas
nth Term T(n) = 1 / [a + (n-1)d]   (where a, d are from the reciprocal AP)
Harmonic Mean HM(a, b) = 2ab / (a + b)
HP Condition If a, b, c are in HP → b = 2ac / (a + c)

AM-GM-HM Relationship and Inequality

The relationship between Arithmetic Mean, Geometric Mean, and Harmonic Mean is one of the most important and versatile concepts in CAT Quant. It appears not only in Progressions questions but also in Inequalities and Algebra problems.

Arithmetic Mean
AM = (a + b) / 2
Geometric Mean
GM = sqrt(a × b)
Harmonic Mean
HM = 2ab / (a + b)
AM   ≥   GM   ≥   HM
Key Identity GM² = AM × HM
Equality condition AM = GM = HM when and only when a = b
Note: The AM-GM inequality is used extensively in CAT to minimise or maximise expressions involving sums and products. When the sum of n positive numbers is fixed, their product is maximum when all are equal — this is the AM-GM equality condition.

Important Sum Formulas

These standard summation results appear directly in CAT questions and are used as intermediate steps in many harder problems. Make sure you can recall each of them instantly.

Sum of Natural Numbers
1+2+3+...+n = n(n+1)/2
Sum of Squares
1²+2²+...+n² = n(n+1)(2n+1)/6
Sum of Cubes
1³+2³+...+n³ = [n(n+1)/2]²
Sum of Odd Numbers
1+3+5+...+(2n-1) = n²
Sum of Even Numbers
2+4+6+...+2n = n(n+1)
GP (3-term shortcut)
Assume 3 terms in GP as: a/r, a, ar
Also Check | CAT Arithmetic Syllabus 2026

Types of Questions Asked in CAT

CAT questions on Progressions and Series can be broadly classified into the following categories. Understanding each type helps you plan which concepts to focus on.

Question Type Description Difficulty
Finding nth Term or Sum Given partial information about an AP or GP, find the nth term or sum of n terms using standard formulae. Easy – Medium 
Recurrence Relations Sequences defined by a formula linking the nth term to previous terms. Requires identifying repeating patterns.  Medium
Common Terms in Two APs  Finding the count of terms common to two APs using LCM of their common differences. Medium
Infinite GP and Series Finding the sum of an infinite GP, or problems involving infinite nested radicals and continued fractions. Hard
AM-GM-HM Inequality Using AM ≥ GM ≥ HM to find minimum or maximum values, or to identify the type of progression. Medium
Mixed AP and GP Problems Sequences where elements satisfy both AP and GP conditions simultaneously. Hard
Real-World Applications Word problems on population growth, compound interest, lab experiments modelled using progressions. Medium
Special Series and Patterns Telescoping series, triangular numbers, sum of n² or n³, and digit-based sequences. Hard

Q1. A bookstore sells an average of 60 books per day during the first 7 days of a new release, and an average of 63 books per day over the first 8 days. On the 9th day, they sell 11 fewer books than on the 8th day. If the daily average from day 2 to day 9 becomes 66 books, exactly how many copies were sold on the very first day?

A.
B.
C.
D. 

Answer: 

Tips and Tricks for CAT Progressions and Series

These expert tips from CAT toppers will help you solve questions faster and avoid the most common mistakes in this topic.

  • Assume smart variables for 3-term AP or GP

    For 3 terms in AP, take them as (a–d), a, (a+d). For 3 terms in GP, take them as a/r, a, ar. This makes the sum and product conditions far simpler to set up and solve.

  • Common terms in two APs form a new AP with d = LCM(d1, d2)

    The first common term is found by inspection. The new common difference of the resulting sequence is the LCM of the two original common differences.

  • For an infinite GP sum to exist, |r| must be strictly less than 1

    Always verify the condition |r| < 1 before applying S∞ = a/(1–r). If |r| ≥ 1, the sum does not converge and the formula cannot be used.

  • Use AM ≥ GM to find minimum or maximum values quickly

    When a problem involves a sum or product of positive quantities under a constraint, the AM-GM inequality almost always gives the extreme value in 2–3 lines without any calculus.

  • Identify the pattern in recurrence sequences early

    For sequences defined by T(n) = f(T(n–1)), compute the first 6–8 terms manually. Most CAT recurrence sequences repeat with a period of 4, 5, or 6. Once you spot the cycle, finding any term is immediate.

  • Telescoping series - look for cancellation patterns

    Series of the form 1/(a1 × a2) + 1/(a2 × a3) + ... can be split using partial fractions. Nearly all terms cancel out, leaving only the first and last terms to evaluate.

  • Sum of n terms as a polynomial in n reveals the progression type

    If S(n) = an² + bn, the sequence is an AP with first term = a + b and common difference = 2a. This is a frequently tested shortcut for backward engineering the AP.

  • In HP problems, always convert to AP by taking reciprocals

    HP questions are rarely solved directly. Take the reciprocal of each term to convert to an AP, solve the standard AP problem, and then convert back. This saves time and avoids errors.

Also Read | CAT Percentage Questions 

How to Prepare Progressions and Series for CAT

A structured preparation approach for this topic will help you score consistently. Follow the steps below to build strong command over the chapter.

  • Build Your Foundation First

    Start with the definitions and standard formulas for AP, GP, and HP. Being able to derive each formula from scratch builds conceptual clarity that pays off in non-standard questions.

  • Master the AM-GM-HM Inequality

    This inequality appears across multiple CAT topics including Progressions, Inequalities, and Algebra. Learn to identify when AM-GM applies and practise bounding problems using it.

  • Solve Year-Wise PYQs Strategically

    Begin with CAT 2018–2020 questions (relatively accessible), then advance to CAT 2021–2025. Observe recurring patterns - the CAT paper-setter tends to revisit similar ideas in different forms.

  • Practise Time-Bound Sets

    Take mini-sectional tests of 5–8 Progressions questions under timed conditions (approximately 12–15 minutes). This builds the habit of prioritising easy vs. hard questions under exam pressure.

  • Integrate with Full-Length Mock Tests

    During full-length CAT mocks, tag Progressions questions and track your accuracy and time separately. This data directly informs your section-wise strategy on exam day.

Check | How to Prepare for CAT 2026: Complete Study Plan and Strategy

CAT Progressions and Series Questions

Below are CAT Previous Year Questions on Progressions and Series from 2025 to 1990, with detailed video solutions by CAT experts. You can also download them as a PDF or attempt them in test format.

Year-wise CAT Progressions and Series questions (2025 to 1990) with detailed solutions will be placed in this section.
This section will be updated shortly.

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Conclusion

CAT Progressions and Series is a high-value topic that rewards consistent preparation. With 5–7 questions appearing every year across all CAT slots, mastering AP, GP, HP, and the AM-GM-HM inequality can significantly boost your Quantitative Aptitude score.

Focus on understanding formulas deeply rather than memorising them, practise year-wise PYQs to recognise question patterns, and integrate this topic into your full-length mock test routine. Candidates who are thorough with this chapter consistently report solving 4–6 questions correctly from this section alone.

Download the free questions PDF and formulas PDF above to get started. For structured coaching, you can also explore the Supergrads CAT Online Course.