Prove the following by the principle of mathematical induction:
1.2 + 2.3 + 3.4 + … + n(n + 1)
Let P(n): 1.2 + 2.3 + 3.4 + … + n(n + 1)=
For n = 1
P(1): 1(1 + 1)=
= 1x2 =
= 2 = 2
Since, P(n) is true for n = 1
Let P(n) is true for n = k
= P(k): 1.2 + 2.3 + 3.4 + … + k(k + 1)= - - - - - (1)
We have to show that,
= 1.2 + 2.3 + 3.4 + … + k(k + 1) + (k + 1)(k + 2)=
Now,
{1.2 + 2.3 + 3.4 + … + k(k + 1)} + (k + 1)(k + 2)
=
= (k + 2)(k + 1)
=
Therefore, P(n) is true for n = k + 1
Hence, P(n) is true for all n ∈ N