Statement - 1: For every natural number n 2, 1 1 + 1 2 + + 1 n > n . Statement -2 : For every natural number n 2, n(n+1) < n+1 .

Step-by-step solution

aligned & P(n)= 1 1 + 1 2 + + 1 n & P(2)= 1 1 + 1 2 > 2 aligned Let us assume that P(k)= 1 1 + 1 2 + + 1 k > k is true P(k+1)= 1 1 + 1 2 + + 1 k + 1 k+1 > k+1 has to be true. L.H.S. > k + 1 k +1 = k ( k +1) +1 k +1 Since k(k+1) >k ( k 0) k ( k +1) +1 k +1 > k +1 k +1 = k +1 Let P(n)= n(n+1) < n+1 Statement -1 is correct. P(2)= 2 3 < 3 If P(k)= k(k+1) < (k+1) is true Now P(k+1)= (k+1)(k+2) < k+2 has to be true Since (k+1) < k+2 ( k +1)( k +2) < ( k +2) Hence Statement -2 is not a correct explanation of Statement -1