Mathematics · Class 11 Science
Ch 8Principle of Mathematical Induction — Class 11 Mathematics, concept-first.
Before stating the principle of mathematical induction formally, it helps to ask a more basic question: what actually characterises the set of natural numbers inside the much larger set of real numbers ?
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Proving Summation Formulas by Induction
Many identities claim that a sum of terms, written using dots such as , equals a closed-form expression in — for example .
Most relevant Q&A
- Prove by induction that $1\cdot2\cdot3 + 2\cdot3\cdot4 + \cdots + n(n+1)(n+2) = \dfrac{n(n+1)(n+2)(n+3)}{4}$ for all $n \ge 1$.Free
- Using the principle of mathematical induction, prove that $1 + 2 + 3 + \cdots + n = \dfrac{n(n+1)}{2}$ for all $n \ge 1$.Free
- Using mathematical induction, prove that $1^2 + 2^2 + 3^2 + \cdots + n^2 = \dfrac{n(n+1)(2n+1)}{6}$ for all $n \ge 1$.Preview
- Prove by induction that $1^3 + 2^3 + 3^3 + \cdots + n^3 = \left[\dfrac{n(n+1)}{2}\right]^2$ for all $n \ge 1$.Free
- Prove that the sum of the first $n$ odd natural numbers is $n^2$, i.e. $1 + 3 + 5 + \cdots + (2n-1) = n^2$, for all $n \ge 1$.Free
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Natural Numbers as the Least Inductive Subset of Real Numbers
Before stating the principle of mathematical induction formally, it helps to ask a more basic question: what actually characterises the set of natural numbers inside the much larger set of real number…
The Principle of Mathematical Induction
Statement of the principle. Let be a statement (a mathematical assertion) involving the natural number . Suppose the following two conditions both hold:
Proving Summation Formulas by Induction
Many series that appear throughout algebra and calculus -- sums of consecutive natural numbers, their squares, their cubes, geometric progressions, and telescoping fractions -- have a compact closed-f…
Proving Divisibility Results by Induction
A second broad family of statements provable by induction asserts that some expression built from is always divisible by a fixed integer, for every natural number .
Proving Inequalities by Induction
Inequality statements behave a little differently from equalities in an induction proof: instead of simplifying both sides down to a single algebraic identity, the inductive step usually chains togeth…
Summary
This chapter developed the principle of mathematical induction, the standard tool for proving a statement true for every natural number (or every from some starting value onward).
More questions
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- Example 1Using the principle of mathematical induction, prove that $1 + 2 + 3 + \cdots + n = \dfrac{n(n+1)}{2}$ for all $n \ge 1$.Free
- Example 2Prove by induction that $7^n - 3^n$ is divisible by $4$ for every natural number $n$.Free
- Example 3Prove that $2^n > n$ for all positive integers $n$.Preview
- Example 4Using mathematical induction, prove that $1^2 + 2^2 + 3^2 + \cdots + n^2 = \dfrac{n(n+1)(2n+1)}{6}$ for all $n \ge 1$.Preview
- Example 5Prove Bernoulli's inequality: $(1+x)^n \ge 1 + nx$ for every natural number $n$, where $x > -1$.Preview
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- Q6Prove by induction that $1^3 + 2^3 + 3^3 + \cdots + n^3 = \left[\dfrac{n(n+1)}{2}\right]^2$ for all $n \ge 1$.Free
- Q7Prove that the sum of the first $n$ odd natural numbers is $n^2$, i.e. $1 + 3 + 5 + \cdots + (2n-1) = n^2$, for all $n \ge 1$.Free
- Q8Prove by induction that $2 + 4 + 6 + \cdots + 2n = n(n+1)$ for all $n \ge 1$.Preview
- Q9Prove that $\dfrac{1}{1\cdot2} + \dfrac{1}{2\cdot3} + \dfrac{1}{3\cdot4} + \cdots + \dfrac{1}{n(n+1)} = \dfrac{n}{n+1}$ for all $n \ge 1$.Preview
- Q10Prove by induction that $1 + 2 + 2^2 + \cdots + 2^{n-1} = 2^n - 1$ for all $n \ge 1$.Preview
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- Q11Prove by induction that $n^3 - n$ is divisible by $6$ for every natural number $n$.Free
- Q12Prove that $n(n+1)(2n+1)$ is divisible by $6$ for every natural number $n$.Free
- Q13Prove by induction that $3^{2n} - 1$ is divisible by $8$ for every natural number $n$.Preview
- Q14Prove that $4^n - 1$ is divisible by $3$ for every natural number $n$.Preview
- Q15Prove by induction that $5^{2n} - 1$ is divisible by $24$ for every natural number $n$.Preview