1² + 3² +5² + …. +(2n –1)² =

P(1) is True
Let us assume P(k) be true
1² + 3² +5² + …. +(2k –1)² =
To prove P(k+1) is true using P(k)
P(k+1) =1² + 3² +5² + …. +(2(k + 1)-1)²  
          =
          = ……………(1)
L.H.S of (1)
1² + 3² +5² + …. +(2(k + 1)-1)² =
                                             =
                                             =
                                             =
                                             =
                                             =
                                             =
which is the R.H.S of ………………………..(1)
P(K+1) is true.
By the Principle of mathematical induction, P(n) is true for all values of n where n N
Hence proved.