Prove that 1 2 + 2 3 + 3 4 + - Maths - Principle of Mathematical Induction

Question

Prove that 1 2 + 2 3 + 3 4 + n(n+1) = 1/3 n (n+1)(n+2)

Aakash Sharma · Accepted Answer

Let the given statement be P(n), i e , P(n):

For n = 1, we have P(1):

, which is true Let P(k) be true for some positive integer k, i e ,

We shall now prove that P(k + 1) is true Consider 1 2 + 2 3 + 3 4 + â€¦ + k (k + 1) + (k + 1) (k + 2) = [1 2 + 2 3 + 3 4 + â€¦ + k (k + 1)] + (k + 1) (k + 2)

Thus, P(k + 1) is true whenever P(k) is true Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i e , n

Prove that 1.2 + 2.3 + 3.4 +..............n(n+1) = 1/3 n (n+1)(n+2)