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Proof by Induction

  2. Algebra
  3. Proof by Induction
    4. 30'

Proof by Induction is a method of proof commonly used in HL mathematics. The method is always the same and questions are worth a good deal of marks in an exam. Therefore, it is really worth investing time to understand how to use it! Questions involving series, divisibility and inequalities are usually fairly straightforward. However, it is also used to make proofs in work on calculus, trigonometry and  complex numbers and these tend to be really quite challenging.

Key Concepts

On this page, you should learn to

  • Carry out proof by induction for a wide variety of topics including
    • series
    • divisibility
    • inequalities
    • differentiation
    • complex numbers
 

Essentials

The following videos will help you understand all the concepts from this page

Proof by Induction for Series

Here is an example of a proof by induction that proves the formula for the sum of cubic numbers:

Prove that \(1^3+2^3+3^3+ ...+n^3=(\frac{n(n+1)}{2})^2 \quad, n\in \mathbb{N}\)

Notes from the video

 

Proof by Induction for Divisibility

Here is a video to show you the method to see if an expression is divisible by a certain quantity

In the following video, we look at an example of a proof by induction for divisibility

Prove that \(n^3+11n\) is divisible by 3 for \(n\in\mathbb{Z} ,n>0\)

Notes from the video

 

Proof by Induction for Inequalities

The following video shows an example of a proof by induction that includes an inequality

Prove that \(n!>2^n\ , \quad n\ge 4\)

Notes from the video

 

 Proof by Induction for Calculus

In the following video we look at an example of proof by induction for differentiation:

Let \(y = \frac{1}{1-x}\) , \(x\in \mathbb{R}\)

Prove by induction that \(\frac{d^ny}{dx^n}=\frac{n!}{(1-x)^{n+1}}\) , \(n\in \mathbb{Z^+}\)

Notes from the video

 

 Proof by Induction for Complex Numbers

In the following video, we look at an example of a proof by induction question applied to complex numbers. In it, we prove De Moivre's Theorem:

Let \(z=r(cosθ+isinθ)\)

Prove that \(z^{ n }≡r^{ n }[cos⁡(nθ)+isin(nθ)]\ ,\ n\in \mathbb{Z^+}\)

Notes from the video

 
 

Summary

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Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

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Exam-style Questions

Question 1

Prove that \(12^n+2\times5^{n-1}\) is divisible by 7 , \(n\in\mathbb{Z^+}\)

Hint

Full Solution

 

Question 2

Prove by induction that \(\sum _{ r=1 }^{ n }{(r\times{ 2 }^{ r-1 })} =(n-1)2^ n+1 \ ,\ n\in\mathbb{Z^+}\)

Hint

Full Solution

 

Question 3

Let \(y = sinx\)

Prove by induction that \(\frac{d^ny}{dx^n}=sin(x+\frac{n\pi}{2})\)

Hint

Full Solution

 

Question 4

Prove by induction that \(sinx+sin3x+sin5x+...+sin(2n-1)x=\frac{1-cos2nx}{2sinx} , \quad n\in\mathbb{Z^+} ,\quad sinx\neq 0\)

Hint

Full Solution

 
 
MY PROGRESS

How much of Proof by Induction have you understood?