In this section 11th grade math, student will learn the new topic prove by mathematical induction.
In algebra or in other disciplines of mathematics, there are certain results or statements that are formulated in terms of 'n' is a positive integer. To prove such statements, well suited method based on the specific technique is known as the Principle of Mathematical Induction.
The Principle of Mathematical Induction
It states that :
i) P(1) is true.
ii) P(k) is true, then P(k+1) is also true, where k is a natural number.
Then the statement P(n) is true for every natural number n.
Examples on Prove by Mathematical Induction
Example 1 : Show that n3 + (n+1)3 + (n+2)3 is divisible by 9 for every natural number 'n'. Solution : P(n) : n3 + (n+1)3 +(n+2)3
First we will verify P(1) is true.
For that , Put n=1 in P(n)
P(1) : 13 + (1+1)3 + ( 1+2)3
P(1) : 1 + 23 + 33
P(1) : 1 + 8 + 27
P(1) : 36
According to the divisibility rule 36 is divisible by 9
∴ P(1) is true . ------(Equation 1)
Let us assume that,
P(k) is true. -------(equation 2)
that means P(k): k3 + (k+1)3+(k+2)3 is divisible by 9.
Now we want to prove that P(k+1) is true.
So P(k+1) will be
P(k+1): (k+1)3 +(k+1+1)3 +(k+1+2)3
P(k+1) : (k+1)3 +(k+2)3 +(k+3)3
P(k+1): (k+1)3+(k+2)3 + k3 + 9k2 +27k + 27
P(k+1):(k+1)3 +(k+2)3 + k3 + 9(k2+3k +3)
From equation (2), (k+1)3+ (k+2)3+k3 is divisible by 9 and 9(k2+ 3k +3)is also divisible by 9.
So, P(k+1) is also divisible by 9. -------(equation 3)
∴ From equation (1), (2) and (3) and by principle of mathematical induction,
n3 + (n+1)3+ (n+2)3 is divisible by 9.
Example 2 : Show that the sum of the first 'n' odd natural numbers is n2. Solution : Odd natural numbers are 1,3,5,... 2n-1
Sum = n2
P(n): 1 + 3 + 5 + ...+(2n-1)=n2
P(1) = 2n-1<
= 2(1) -1
= 2 -1 =1
n2 = (1)2
∴ P(1) is true. ----(equation 1)
Assume that P(k) is true.
So P(k): 1 + 3 + 5 +....(2k -1) = k2 is true. ----(equation 2)
Now we have to prove that P(k+1) is true.
P(k+1) : 1 + 3 + 5 +....+(2k-1) + [2(k+1)-1]
= k2 + (2k + 1)
∴ P(k+1) is true whenever p(k) is true.
Hence by equation (1), (2) and by principle of mathematical induction P(n) is true for all natural number 'n'.
We at ask-math believe that educational material should be free for everyone. Please use the content of this website for in-depth understanding of the concepts. Additionally, we have created and posted videos on our youtube.
We also offer One to One / Group Tutoring sessions / Homework help for Mathematics from Grade 4th to 12th for algebra, geometry, trigonometry, pre-calculus, and calculus for US, UK, Europe, South east Asia and UAE students.
Affiliations with Schools & Educational institutions are also welcome.
Please reach out to us on [email protected] / Whatsapp +919998367796 / Skype id: anitagovilkar.abhijit
We will be happy to post videos as per your requirements also. Do write to us.