MathDB

Let

\left( a_n \right)_{n\ge 1}

be a sequence of real numbers such that

a_1>2

and

a_{n+1} =a_1+\frac{2}{a_n} ,

for all natural numbers

n.

a) Show that

a_{2n-1} +a_{2n} >4 ,

for all natural numbers

n,

and

\lim_{n\to\infty} a_n =2.

b) Find the biggest real number

a

for which the following inequality is true: \sqrt{x^2+a_1^2} +\sqrt{x^2+a_2^2} +\sqrt{x^2+a_3^2} +\cdots +\sqrt{x^2+a_n^2} > n\sqrt{x^2+a^2}, \forall x\in\mathbb{R} , \forall n\in\mathbb{N} .

Problem Statement