​ For all n ≥ 1 , prove that 1 + 2 + 3 +…….+ n < 1/8 (2n + 1) ​

Question

For all n ≥ 1 , prove that 1 + 2 + 3 +…….+ n < \(\frac{1}{8}\) (2n + 1)

Answer 1

Let p(n): 1 + 2 + 3 +…….+ n , Put n = 1 ⇒ p(1) = 1 < \(\frac{9}{8}\) which is true.

Assuming that true for p(k)

p(k): 1 + 2 + 3 +…….+ k < \(\frac{1}{8}\)(2k + 1)²

Let p(k +1): 1 + 2 + 3 +……..+ k + (k +1) < \(\frac{1}{8}\)

(2k + 1)² + (k + 1)

Hence by using the principle of 

mathematical induction true for all n ∈ N.