CAT 2026 Arithmetic Progression and Geometric Progression: Formulas, Questions and Shortcuts

CAT 2026 Progressions and Series: Important Formulas

Before attempting CAT algebra questions on sequences and series, revise these formulas. They are useful for finding individual terms, sums and common differences.

Here, a is the first term, d is the common difference, r is the common ratio and l is the last term.

CAT 2026 AP Questions From Previous Year Papers

AP questions often provide relationships between selected terms rather than giving the first term and common difference directly. The following examples from the CAT previous year questions illustrate this approach.

Question 1: CAT 2024 Slot 1 - Finding the 100th Term of an AP

The terms x1,x2,x3,,x100 are in an arithmetic progression. If x5=-4 and 2x6+2x9=x11+x13, find x100.

Solution:

Let the first term be a and the common difference be d.

Given:

x5=a+4d=-4

Using the second condition:

2(a+5d)+2(a+8d)=(a+10d)+(a+12d)

Simplifying:

4a+26d=2a+22d

2a+4d=0

Therefore, a=-2d.

Substitute this into a+4d=-4:

-2d+4d=-4

d=-2, so a=4.

Now,

x100=a+99d

x100=4+99(-2)=-194

Answer: -194 (Option A).

CAT takeaway: Convert each term into a+(n-1)d before using the given relationship. This keeps the equation organised and reduces calculation errors.

Question 2: CAT 2022 - Finding the Smallest Value of (n)

The sum of the first n terms of an arithmetic progression is given by Sn=n+2n2. If the nth term is divisible by 9, find the smallest possible positive value of n.

Solution:

The nth term is the difference between two consecutive partial sums:

an=Sn-Sn-1

Given:

Sn=n+2n2

Sn-1=(n-1)+2(n-1)2

Therefore,

an=(n+2n2)-[(n-1)+2(n-1)2]

Simplifying:

an=4n-1

Since the term must be divisible by 9:

$4n-1\equiv0\pmod9$

$4n\equiv1\pmod9$

The smallest positive integer satisfying this condition is n=7, because 4(7)-1=27.

Answer: 7 (Option C).

CAT takeaway: When the sum of the first n terms is provided, calculate an using Sn-Sn-1 instead of trying to identify the first term and common difference separately.

Question 3: CAT 2023 Slot 2 - Finding a Term From Two APs

Two arithmetic progressions, a1,a2, and b1,b2,, have prime-number common differences. Given that a5=b9, a19=b19 and b2=0, find a11.

Solution:

Let the first terms be A and B, and the common differences be d and $e.

Since b2=0:

B+e=0B=-e

Using a5=b9:

A+4d=B+8e=7e

Using a19=b19:

A+18d=B+18e=17e

Subtract the first equation from the second:

14d=10e

7d=5e

Since both d and e are prime numbers, d=5 and e=7.

From A+4d=7e:

A+20=49A=29

Therefore,

a11=A+10d=29+50=79

Answer: 79 (Option A).

CAT takeaway: When two APs are involved, use their term formulas together. Conditions involving prime common differences can help determine the exact values of the differences.

CAT 2026 GP Questions and Sum of Infinite Series

GP questions can require candidates to identify the common ratio, use relationships between consecutive terms or determine whether an infinite series has a finite sum.

Question 4: CAT 2025 Slot 2 - Sum of a Decreasing Infinite GP

The terms of a decreasing infinite geometric progression are a1,a2,a3,. If

a1+a2+a3=52

and

a1a2+a2a3+a3a1=624,

find the sum of the infinite progression.

Solution:

Let the first term be a and the common ratio be r.

Then:

a1+a2+a3=a(1+r+r2)=52

Also,

a1a2+a2a3+a3a1=a2r(1+r+r2)=624

Divide the second equation by the first:

ar=62452=12

Thus, a=12r.

Substitute into a(1+r+r2)=52:

12r(1+r+r2)=52

Simplifying:

3r2-10r+3=0

Factoring:

(3r-1)(r-3)=0

Therefore, r=13 or r=3.

For a decreasing infinite GP with a finite sum, |r|<1. Hence, r=13.

Since ar=12, the first term is a=36.

The sum of the infinite GP is:

S=a1-r

S=361-13=54

Answer: 54 (Option A).

CAT takeaway: In GP questions involving both sums and products of terms, dividing the given equations can eliminate the repeated factor and reveal the common ratio. Always check the condition |r|<1 before using the infinite-sum formula.

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CAT Progressions and Series: Additional PYQ Practice

Practise these CAT 2021-2025 PYQs to strengthen your understanding of AP, GP, recurrence relations, and sequence-based algebra. These questions test your ability to apply progression formulas to numerical, geometrical, and algebraic problems.

These questions show why preparation should go beyond memorising formulas. CAT can test progressions through number relationships, geometry, algebraic conditions and recurrence-based sequences.

Shortcuts to Solve AP and GP Questions Faster

These shortcuts help you identify common patterns in AP and GP questions and solve them without calculating every term.

These methods are especially useful when a CAT question gives only a few terms or sums and asks you to find a distant term or the value of the series.

How to Prepare for CAT 2026 Progressions and Series?

• Revise AP first: Practise finding the common difference and using relationships between non-consecutive terms.

• Move to GP: Focus on the common ratio, finite sums and conditions for an infinite sum.

• Practise mixed conditions: Work on questions involving two progressions, sums of terms and algebraic constraints.

• Review mistakes: Reattempt difficult PYQs after a few days and note whether you can solve them faster.

• Use timed practice: Focus on choosing an efficient approach instead of spending too long on one question.

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