Question -
Answer -
Let P (n): 1.2 + 2.3 + 3.4 + … + n(n+1) = [n (n+1)(n+2)] / 3
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.
Now, let us check for P (n) is true for n = k, and have toprove that P (k + 1) is true.
P (k): 1.2 + 2.3 + 3.4 + … + k(k+1) = [k (k+1) (k+2)] / 3 …(i)
So,
1.2 + 2.3 + 3.4 + … + k(k+1) + (k+1) (k+2)
Now, 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
Hence, P (n) is true for all n ∈ N.