MATHEMATICAL INDUCTION EXAMPLES

About "Mathematical Induction Examples"

Mathematical Induction Examples :

Here we are going to see some mathematical induction problems with solutions.

Define mathematical induction :

Mathematical Induction is a method or technique of proving mathematical results or theorems

The process of induction involves the following steps.

Mathematical Induction Examples

Question 1 :

By the principle of mathematical induction, prove that, for n ≥ 1

1² + 3² + 5² + · · · + (2n − 1)² = n(2n − 1)(2n + 1)/3

Solution :

Let p(n) =  1² + 3² + 5² + · · · + (2n − 1)² = n(2n − 1)(2n+1)/3

Step 1 :

put n = 1

p(1)  = 1² + 3² + 5² + · · · + (2(1) − 1)² = 1(2(1) − 1)(2(1)+1)/3

  1  =  1

Hence p(1) is true.

Step 2 :

Let us assume that the statement is true for n = k

p(k) = 1²+3²+5²+ · · · + (2k − 1)² = k(2k − 1)(2k+1)/3  ---(1)

We need to show that P(k + 1) is true. Consider,

Step 3 :

Let us assume that the statement is true for n = k + 1

p(k+1) 

1² + 3² · · · + (2(k+1) − 1)² = [(k+1)(2(k+1) − 1)(2(k+1)+1)]/3 

1² + 3² · · · + (2k+1)² = [(k+1)(2k + 1) (2k + 3)]/3 

1² + 3² · · · + (2k-1)² + (2k+1)² = [(k+1)(2k + 1) (2k + 3)]/3 

By applying (1) in this step, we get

Hence, by the principle of mathematical induction,n ≥ 1

1² + 3² + 5² + · · · + (2n − 1)² = n(2n − 1)(2n + 1)/3

Question 2 :

Prove that the sum of the first n non-zero even numbers is n² + n.

Solution :

Let p(n) be the statement "n² + n" is even.

Step 1 :

p(n)  =  n² + n

put n = 1

p(1)  =  1² + 1  =  2, which is even

Hence p(1) is true.

Ste 2 :

Let p(m) is true. Then 

p(m) is true  ==>  m² + m is even ==>  m² + m  ==> 2λ for some λ ∊ N

Now, we shall show that p(m + 1) is true. For this we have to show that (m+1)² + (m + 1) is an even natural number.

Now,

(m+1)² + (m + 1)  =  m² + 2 m + 1 + m + 1

  =  m² + 2 m + m + 2

  =  m² + m + 2m + 2

  =  m² + m + 2(m + 1)

  =  2λ + 2 (m + 1)

  =  2(λ + m + 1)

Hence p(m + 1) is true.

Hence the sum of the first n non-zero even numbers is n² + n.

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