Suppose P (n): 1.2 + 2.3 + 3.4 + … + n(n + 1) = [n(n + 1) (n + 2)]/3
Now let us check for n = 1,
P (1): 1(1 + 1) = [1(1 + 1) (1 + 2)]/3
: 2 = 2
P (n) is true for n = 1.
Then, let us check for P (n) is true for n = k, and have to prove that P (k + 1) is true.
P (k): 1.2 + 2.3 + 3.4 + … + k(k + 1) = [k(k + 1) (k + 2)]/3 … (i)
Therefore,
1.2 + 2.3 + 3.4 + … + k(k + 1) + (k + 1) (k + 2)
Then, substituting the value of P (k) we get,
= [k(k + 1) (k + 2)]/3 + (k + 1) (k + 2) by using equation (i)
= (k + 2) (k + 1) [k/2 + 1]
= [(k + 1) (k + 2) (k + 3)]/3
P (n) is true for n = k + 1
Thus, P (n) is true for all n ∈ N.
