BALTIC-WAY 2005 Problem 3

Consider the sequence $a_{k}$ defined by $a_{1}=1, a_{2}=\frac{1}{2}$,

$$a_{k+2}=a_{k}+\frac{1}{2} a_{k+1}+\frac{1}{4 a_{k} a_{k+1}} \text { for } k \geq 1$$

Prove that

$$\frac{1}{a_{1} a_{3}}+\frac{1}{a_{2} a_{4}}+\frac{1}{a_{3} a_{5}}+\cdots+\frac{1}{a_{98} a_{100}}<4$$