1 + 2+ 3 +…. + n < ( 2n + 1)²

Let P(n) = 1 + 2+ 3 +…. + n < ( 2n + 1)²
P(1) is true
Let us assume p(k) is true.
1 + 2+ 3 +…. + k < ( 2k + 1)²
To prove P(k+1) is true using P(k)
P(k+1) = 1 + 2+ 3 +…. + k +k+1< ( 2(k + 1)+1)²
            = 1+ (1 + 2+ 3 +…. + k) +k < ( 2k + 3)²…….. 1
L .H.S 1+ ( 2k + 1)² + k
          =
            = ( 4k² + 1 +4k + 8k + 8)
            = (4k² + 12k +9)
            = (2k + 3)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.