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 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 \(1^3+2^3+3^3+ ...+n^3=(\frac{n(n+1)}{2})^2 \quad, n\in \mathbb{N}\)
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 \(n\in\mathbb{Z} ,n>0\)
Proof by Induction for Inequalities
The following video shows an example of a proof by induction that includes an inequality
Prove that \(n!>2^n\ , \quad n\ge 4\)
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 \(\frac{d^ny}{dx^n}=\frac{n!}{(1-x)^{n+1}}\) , \(n\in \mathbb{Z^+}\)
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 \(z^{ n }≡r^{ n }[cos(nθ)+isin(nθ)]\ ,\ n\in \mathbb{Z^+}\)
Here is a quiz that practises the skills from this page
Put the 5 steps to a proof by induction in the correct order
Let P(n) be the proposition... Show true for n=1 Assume true for n=k Show true for n=k+1 Concluding statement
Fill in the gaps to the following proof by induction
2k-1 (k+1)² Assume Show n=k+1 2k+1 RHS LHS
Complete the gaps in the following proof by induction
Let P(n) be the proposition that 9n - 1 is divisible by 8 , \(n\in \mathbb{N}\)
Show true for n=1,
91 - 1 = 8 , 8 is divisible by 8
Hence true for n=1
Assume true for n=k
\(\frac{9^k-1}{8}=m\) , \(n\in \mathbb{Z}\)
\(9^k\) =
Show true for n=k+1
9k+1 - 1= k - 1
9k+1 - 1=9(8m+1)-1
9k+1 - 1=72m+
9k+1 - 1=8(
Hence 9k+1 - 1 is divisible by 8
True for n=k+1
Concluding statement
True for n=1
Assuming it is true for n=k then it is true for n=
Therefore it is true for all n , \(n\in \mathbb{N}\)
Prove that \(12^n+2\times5^{n-1}\) is divisible by 7 , \(n\in\mathbb{Z^+}\)
For assumption n = k is true
\(12^k+2\times5^{k-1}=7m\) , m is an integer
Prove by induction that \(\sum _{ r=1 }^{ n }{(r\times{ 2 }^{ r-1 })} =(n-1)2^ n+1 \ ,\ n\in\mathbb{Z^+}\)
You might find it easier to if you write out the series in full:
\(\sum _{ r=1 }^{ n }{(r\times{ 2 }^{ r-1 })} =1\times2^0+3\times2^1+1\times2^2+...+n\times{ 2 }^{ n-1 }\)
Let \(y = sinx\)
Prove by induction that \(\frac{d^ny}{dx^n}=sin(x+\frac{n\pi}{2})\)
The following trigonometric identity is useful for this proof:
\(cosx \equiv sin(x+\frac{\pi}{2})\)
Prove by induction that \(sinx+sin3x+sin5x+...+sin(2n-1)x=\frac{1-cos2nx}{2sinx} , \quad n\in\mathbb{Z^+} ,\quad sinx\neq 0\)
This is a challenging proof!
The following identities will be useful:
\(sin2x \equiv 2sinxcosx \\ cos2x \equiv 1-sin^2x\)
\(sin(A+B) \equiv sinAcosB+cosAsinB\\ sin(2kx+x) \equiv sin2kxcosx+cos2kxsinx\)
\(cos(A+B) \equiv cosAcosB+sinAsinB\\ cos(2kx+2x) \equiv cos2kxcos2x+sin2kxsin2x\)
MY PROGRESS
Self-assessment How much of Proof by Induction have you understood?
My notes
Which of the following best describes your feedback?
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.
Twitter 
Facebook 
LinkedIn