Solve:12 + 32 + 52 + … + (2n – 1)2 = 1/3 n (4n2 – 1)

Let ,

P(n) be the given equation,

Let us check for

   

   

   

P (n) is true for

   

Now, let us check for P (n) is true for

   

, and have to prove that P (k + 1) is true.

 

   

So,

   

Now, substituting the value of P (k) we get,

   

by using equation (i)

   

or,

   

or,

   

or,

   

   

or,

   

   

or,

   

   

or,

   

P (n) is true for

   

Hence, P (n) is true for all n ∈ N