proving Inequality

think jensen...:)

It is not jensen

Let P(n):√1+√2+√3....√n≥√n(n+1)/2

P(1):√1≥√1*2/2
Hence P(1) is true

Let P(n) be true
Before going to P(n+1) we note that-

(√n+1)²=n+1

while (√n+√2)²=n+2+2√2n

therefore √n+√2≥√n+2

P(n+1):√1+√2+√3+...√n+1 ≥ √n(n+1)2+√n+1=√(n+1)2*{√n+√2]
≥√(n+1)(n+2)2

Which gives us the required result.
Hence P(n) is true for all natural numbers.