Prove the following by the principle of mathematical induction:
1.3 + 2.4 + 3.5 + … + n . (n + 2)
Let P(n): 1.3 + 2.4 + 3.5 + … + n.(n + 2) =
For n = 1
P(1): 1.3 = .1.(2)(9)
= 3 = 3
Since, P(n) is true for n = 1
Now,
For n = k
= P(n): 1.3 + 2.4 + 3.5 + … + k . (k + 2)= - - - - - (1)
We have to show that
= 1.3 + 2.4 + 3.5 + … + k . (k + 2) + (k + 3) =
Now,
= {1.3 + 2.4 + 3.5 + … + k (k + 2)} + (k + 1)(k + 3)
= k(k + 1)(2k + 7) + (k + 1)(k + 3) using equation (1)
= (k + 1)
= (k + 1)
= (k + 1)
= (k + 1)
= (k + 1)
= (k + 1)
= (k + 1)(k + 2)(2k + 9)
Therefore, P(n) is true for n = k + 1
Hence, P(n) is true for all n∈ N