Mathematical Induction formulas for JEE Main & Advanced

15 formula entries for Mathematical Induction, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Proving Divisibility Using Induction

f (k)

is divisible by

d

, then

f (k + 1) = f (k) + multiple of d

When to use. Use when the statement involves divisibility by a fixed integer (e.g., $3^{2 n} - 1$ is divisible by 8).

Trap. Not explicitly factoring out d from the expression for f(k+1).

Proving Inequalities Using Induction

Example: Prove

2^{n} > n

for

n \geq 1

. Assume

2^{k} > k

, then

2^{k + 1} = 2 \cdot 2^{k} > 2 k \geq k + 1

for

k \geq 1

When to use. Use when proving statements like $n! > 2^{n}$ for $n \geq 4$ or $2^{n} > n^{2}$ for $n \geq 5$ .

Trap. Forgetting to check the base case for the smallest n in the inequality.

Proving Summation Formulas Using Induction

S (k) = \frac{k (k + 1)}{2}

, then

S (k + 1) = S (k) + (k + 1) = \frac{k (k + 1)}{2} + (k + 1) = \frac{(k + 1) (k + 2)}{2}

When to use. Use for proving formulas for sums of arithmetic, geometric, or other series (e.g., $\sum_{i = 1}^{n} i^{2} = \frac{n (n + 1) (2 n + 1)}{6}$ ).

Trap. Miscalculating the (k+1)-th term, especially in series with alternating signs or fractions.

Handling Base Cases Other Than n=1

Example: Prove

2^{n} > n^{2}

for

n \geq 5

. Base case:

2^{5} = 32 > 25

holds.

When to use. Use when the statement is only true for n ≥ some integer (e.g., inequalities that fail for small n).

Trap. Using n=1 as base case when the statement is false for small n.

Using Induction with Binomial Theorem

Example: Prove

3^{2 n} - 1

divisible by 8. Write

3^{2 (k + 1)} - 1 = 9 \cdot 3^{2 k} - 1 = 9 (3^{2 k} - 1) + 8

When to use. Use when the expression involves powers like $a^{k n}$ or $(a + b)^{n}$ and you need to prove divisibility or inequality.

Trap. Not simplifying the expression to isolate the inductive hypothesis term.

Proving Statements About Sequences

Example: For

a_{1} = 1

a_{n + 1} = 2 a_{n} + 1

, prove

a_{n} = 2^{n} - 1

. Inductive step:

a_{k + 1} = 2 (2^{k} - 1) + 1 = 2^{k + 1} - 1

When to use. Use when a sequence is defined recursively and you need to prove a formula or property.

Trap. Not using the recurrence relation correctly in the inductive step.

Induction on Sets or Combinatorial Statements

Example: Prove a set with n elements has

2^{n}

subsets. Base: n=0 gives 1 subset. Inductive: add one element, doubling the number.

When to use. Use for combinatorial proofs involving finite sets, like counting subsets, permutations, or combinations.

Trap. Not handling the empty set correctly in the base case.

Using Induction with Trigonometric Identities

Example: Prove

\sum_{r = 1}^{n} \sin (r θ) = \frac{\sin (\frac{n θ}{2}) \sin (\frac{(n + 1) θ}{2})}{\sin (\frac{θ}{2})}

When to use. Use when proving formulas for sums of sines, cosines, or other trigonometric series.

Trap. Not using correct trigonometric identities to simplify the sum.

Proving Statements About Polynomials

Example: Prove

n^{3} - n

is divisible by 3. Inductive:

(k + 1)^{3} - (k + 1) = (k^{3} - k) + 3 k (k + 1)

When to use. Use for polynomial expressions like $n^{5} - n$ divisible by 5, or sums of powers.

Trap. Not expanding correctly to isolate the inductive hypothesis.

Handling Inequalities with Factorials

Example:

(k + 1)! > (k + 1) 2^{k} \geq 2 \cdot 2^{k} = 2^{k + 1}

k + 1 \geq 2

When to use. Use for inequalities involving factorials, exponentials, or powers, especially when the base case is not n=1.

Trap. Not checking the condition for the inequality to hold (e.g., $k + 1 \geq 2$ is always true, but for $k + 1 \geq 4$ it's different).

Induction with Multiple Variables

Example: Prove

a^{m + n} = a^{m} a^{n}

for natural numbers m,n. Fix m, induct on n.

When to use. Use when the statement is symmetric or involves two independent natural numbers.

Trap. Not clearly stating which variable is being inducted upon.

Using Induction to Prove Properties of Determinants

Example: Prove determinant of an n×n matrix with 2 on diagonal and 1 on off-diagonals equals n+1.

When to use. Use for determinants that follow a recurrence relation, like those in JEE problems on special matrices.

Trap. Not correctly expanding the determinant to relate to smaller cases.

Induction with Complex Numbers

De Moivre:

(\cos θ + i \sin θ)^{n} = \cos (n θ) + i \sin (n θ)

. Inductive step: multiply by

\cos θ + i \sin θ

When to use. Use for proving properties of complex numbers that involve natural number exponents.

Trap. Not handling the case when n is negative or fractional (only natural numbers).

Induction on the Number of Terms in a Product

Example: Prove

\prod_{r = 1}^{n} (1 + x_{r}) \geq 1 + \sum_{r = 1}^{n} x_{r}

when all

x_{r} \geq 0

or all

x_{r} \in (- 1, 0]

(same sign, each

> - 1

When to use. Use for statements about products, like inequalities or divisibility of products.

Trap. Not using the inductive hypothesis correctly when multiplying.

Proving Bernoulli's Inequality Using Induction

(1 + x)^{n} \geq 1 + n x

for

x > - 1

n \in N

When to use. Use when proving inequalities involving $(1 + x)^{n}$ or in limit problems.

Trap. Forgetting the condition $x > - 1$ to ensure $(1 + x) > 0$ .

Read the full chapter: Mathematical Induction for JEE, with what JEE asks and the common traps.