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Mathematics · Class 11 Science

Ch 2Principle of Mathematical Induction — Class 11 Mathematics, concept-first.

Mathematics rests on two complementary styles of reasoning: deduction and induction.

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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 .

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Chapter contents

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4.1

Introduction

Mathematics rests on two complementary styles of reasoning: deduction and induction.

4.2

Motivation

Imagine a long row of thin rectangular tiles standing on their edges, close enough together that if one tips it knocks over its neighbour.

4.3

The Principle of Mathematical Induction

Let be a mathematical statement involving a natural number . The Principle of Mathematical Induction says is true for every natural number provided both of the following hold:

+Worked Examplesi8 questions
  1. Example 1For all $n \ge 1$, prove that $1^2 + 2^2 + 3^2 + 4^2 + \ldots + n^2 = \dfrac{n(n+1)(2n+1)}{6}$.Free
  2. Example 2Prove that $2^n > n$ for all positive integers $n$.Free
  3. Example 3For all $n \ge 1$, prove that $\dfrac{1}{1.2} + \dfrac{1}{2.3} + \dfrac{1}{3.4} + \ldots + \dfrac{1}{n(n+1)} = \dfrac{n}{n+1}$.Free
  4. Example 4For every positive integer $n$, prove that $7^n - 3^n$ is divisible by 4.Preview
  5. Example 5Prove that $(1 + x)^n \ge (1 + nx)$, for all natural number $n$, where $x > -1$.Preview
  6. Example 6Prove that $2.7^n + 3.5^n - 5$ is divisible by 24, for all $n \in N$.Preview
  7. Example 7Prove that $1^2 + 2^2 + \ldots + n^2 > \dfrac{n^3}{3}$, $n \in N$.Preview
  8. Example 8Prove the rule of exponents $(ab)^n = a^n b^n$ by using principle of mathematical induction for every natural number.Preview

Exercises

+Show 24 questions24 questions
  1. Q1Prove the following by using the principle of mathematical induction for all $n \in N$: $1 + 3 + 3^2 + \ldots + 3^{n-1} = \dfrac{3^n - 1}{2}…Free
  2. Q2Prove the following by using the principle of mathematical induction for all $n \in N$: $1^3 + 2^3 + 3^3 + \ldots + n^3 = \left(\dfrac{n(n+1…Free
  3. Q3Prove the following by using the principle of mathematical induction for all $n \in N$: $1 + \dfrac{1}{1+2} + \dfrac{1}{1+2+3} + \ldots + \d…Free
  4. Q4Prove the following by using the principle of mathematical induction for all $n \in N$: $1.2.3 + 2.3.4 + \ldots + n(n+1)(n+2) = \dfrac{n(n+1…Preview
  5. Q5Prove the following by using the principle of mathematical induction for all $n \in N$: $1.3 + 2.3^2 + 3.3^3 + \ldots + n.3^n = \dfrac{(2n-1…Preview
  6. Q6Prove the following by using the principle of mathematical induction for all $n \in N$: $1.2 + 2.3 + 3.4 + \ldots + n.(n+1) = \left[\dfrac{n…Preview
  7. Q7Prove the following by using the principle of mathematical induction for all $n \in N$: $1.3 + 3.5 + 5.7 + \ldots + (2n-1)(2n+1) = \dfrac{n(…Preview
  8. Q8Prove the following by using the principle of mathematical induction for all $n \in N$: $1.2 + 2.2^2 + 3.2^3 + \ldots + n.2^n = (n-1)2^{n+1}…Preview
  9. Q9Prove the following by using the principle of mathematical induction for all $n \in N$: $\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \ldots…Preview
  10. Q10Prove the following by using the principle of mathematical induction for all $n \in N$: $\dfrac{1}{2.5} + \dfrac{1}{5.8} + \dfrac{1}{8.11} +…Preview
  11. Q11Prove the following by using the principle of mathematical induction for all $n \in N$: $\dfrac{1}{1.2.3} + \dfrac{1}{2.3.4} + \dfrac{1}{3.4…Preview
  12. Q12Prove the following by using the principle of mathematical induction for all $n \in N$: $a + ar + ar^2 + \ldots + ar^{n-1} = \dfrac{a(r^n -…Preview
  13. Q13Prove the following by using the principle of mathematical induction for all $n \in N$: $\left(1+\dfrac{3}{1}\right)\left(1+\dfrac{5}{4}\rig…Preview
  14. Q14Prove the following by using the principle of mathematical induction for all $n \in N$: $\left(1+\dfrac{1}{1}\right)\left(1+\dfrac{1}{2}\rig…Preview
  15. Q15Prove the following by using the principle of mathematical induction for all $n \in N$: $1^2 + 3^2 + 5^2 + \ldots + (2n-1)^2 = \dfrac{n(2n-1…Preview
  16. Q16Prove the following by using the principle of mathematical induction for all $n \in N$: $\dfrac{1}{1.4} + \dfrac{1}{4.7} + \dfrac{1}{7.10} +…Preview
  17. Q17Prove the following by using the principle of mathematical induction for all $n \in N$: $\dfrac{1}{3.5} + \dfrac{1}{5.7} + \dfrac{1}{7.9} +…Preview
  18. Q18Prove the following by using the principle of mathematical induction for all $n \in N$: $1 + 2 + 3 + \ldots + n < \dfrac{1}{8}(2n + 1)^2$Preview
  19. Q19Prove the following by using the principle of mathematical induction for all $n \in N$: $n (n + 1) (n + 5)$ is a multiple of 3.Preview
  20. Q20Prove the following by using the principle of mathematical induction for all $n \in N$: $10^{2n - 1} + 1$ is divisible by 11.Preview
  21. Q21Prove the following by using the principle of mathematical induction for all $n \in N$: $x^{2n} - y^{2n}$ is divisible by $x + y$.Preview
  22. Q22Prove the following by using the principle of mathematical induction for all $n \in N$: $3^{2n+2} - 8n - 9$ is divisible by 8.Preview
  23. Q23Prove the following by using the principle of mathematical induction for all $n \in N$: $41^n - 14^n$ is a multiple of 27.Preview
  24. Q24Prove the following by using the principle of mathematical induction for all $n \in N$: $(2n + 7) < (n + 3)^2$.Preview