Mathematical Induction, Binomial Theorem- Must Do Questions ISI–CMI Maths 2026

Preparing for ISI–CMI Maths 2026 requires strong clarity in core concepts. Many students study multiple topics, but some areas carry more importance than others. Among them, mathematical induction plays a major role. The binomial theorem also supports many problems, but induction remains the primary focus.

Students often feel confused about how to approach such questions. Some try to memorise formulas. Others solve questions without understanding the logic. This approach does not work in exams like ISI and CMI. These exams test your thinking, not just memory.

Here, we will explain mathematical induction and the binomial theorem in simple language. We will also cover important question types that you must practice for ISI–CMI Maths 2026.

Why Mathematical Induction is Important for ISI–CMI 2026

Mathematical induction is a method used to prove statements for all natural numbers. It is very useful in problems involving sequences, divisibility, and inequalities.

This topic is important because:

• Many questions are based on pattern-based proofs

• It helps in building logical thinking

• It is useful in number theory and algebra

• It improves structured problem-solving

If you understand induction clearly, you can solve a wide range of questions with confidence.

What is Mathematical Induction?

Mathematical induction is a step-by-step proof method. It helps in proving that a statement is true for all values starting from a certain number.

It works in three steps:

1. Base Case

You check whether the statement is true for the starting value.
This is usually n=1, n=0, or sometimes n=3.

2. Inductive Hypothesis

You assume that the statement is true for some value n=k.

3. Inductive Step

You prove that the statement is also true for n=k+1.

If all three steps are correct, the statement is true for all natural numbers.

Standard Formulas You Should Know

These formulas are often used in induction problems:

You should not just memorise them. Try to prove them using induction at least once.

Must Do Question Type 1: Divisibility Using Induction

One common question type involves proving divisibility.

Example Concept

You may be given a function like:

Approach

• Start by checking small values

• Use algebraic manipulation

• Break expressions using identities

• Apply induction carefully

These questions test your ability to combine algebra with logic.

Must Do Question Type 2: Existence Problems

These questions ask whether a certain set of numbers exists.

Example Concept

For every n≥3, find nnn distinct positive integers such that each number divides the sum of the remaining numbers.

Key Idea

• Define the sum of all numbers

• Use divisibility properties

• Apply induction to move from n=k to n=k+1

These problems may look difficult at first, but they follow a clear pattern once you understand the logic.

Must Do Question Type 3: Number Construction Problems

These questions involve constructing numbers with specific properties.

Example Concept

For every positive integer nnn, prove that there exists an nnn-digit number divisible by 5n, and all digits are odd.

Approach

• Start with a base case like n=1

• Assume a number exists for n=k

• Extend it to n=k+1

• Use modular arithmetic

This type of question tests creativity along with induction.

Must Do Question Type 4: Perfect Square Problems

These questions involve identifying when an expression becomes a perfect square.

Example Concept

Find all values of nnn such that:

is a perfect square.

Approach

• Assume the expression equals b2

• Use inequalities to bound values

• Apply approximation techniques

• Check small values directly

In many cases, only a few values satisfy the condition.

Must Do Question Type 5: Pairwise Relatively Prime Numbers

These are advanced problems, but very important.

Example Concept

Find numbers k0,k1,...,kn​ such that:

• All numbers are greater than 1

• They are pairwise relatively prime

• Their product minus one becomes the product of two consecutive integers

Approach

• Start with a small base case

• Use induction to extend

• Apply the gcd properties

• Carefully construct the next term

These questions test a deep understanding of number theory.

Introduction to Binomial Theorem

The binomial theorem helps in expanding expressions of the form:

Role of Binomial Theorem in ISI–CMI

The binomial theorem is not heavily asked directly. But it is used in:

• Simplifying expressions

• Expanding terms in proofs

• Solving inequalities

• Supporting induction steps

So, you should understand it clearly, even if it is not the main focus.

Preparation Strategy for ISI–CMI Maths 2026

A clear strategy can improve your performance.

Focus on Concepts

Do not rush through topics. Understand each concept properly.

Practice Regularly

Solve different types of questions. Try Olympiad-level problems.

Write Full Proofs

Do not just think. Write complete solutions step by step.

Revise Frequently

Go back to important questions again and again.

Analyse Mistakes

Understand where you went wrong and improve.

Mathematical induction is a powerful tool. It helps in solving many types of problems in ISI–CMI Maths 2026. The binomial theorem supports your understanding of algebra and expansions.

You do not need to study everything at once. Focus on clarity and consistency. Start with simple problems. Then move to the advanced ones.

With regular practice and the right approach, you can improve your problem-solving skills and handle these questions with confidence.