MathDB

Consider the sequence

(a_n)_{n\ge 1}

such that

a_1=1

and

a_{n+1}=\sqrt{a_n+n^2}

\forall n\ge 1

<span class='latex-bold'>(a)</span>

Prove that there is exactly one rational number among the numbers

a_1,a_2,a_3,\dots

<span class='latex-bold'>(b)</span>

Consider the sequence

(S_n)_{n\ge 1}

such that

S_n=\sum_{i=1}^n\frac{4}{\left (\left \lfloor a_{i+1}^2\right \rfloor-\left \lfloor a_i^2\right \rfloor\right)\left(\left \lfloor a_{i+2}^2\right \rfloor-\left \lfloor a_{i+1}^2\right \rfloor\right)}.

Prove that there exists an integer

N

such that

S_n>0.9

\forall n>N

.
(Stefan Obadă)

Problem Statement