Find the limit $lim_{nto infty }frac{1}{a_1a_2}+frac{1}{a_2a_3}+cdots+frac{1}{a_na_{n-1}}$

Given $a_1=1$ and $a_n=a_{n-1}+4$ where $n\geq2$ calculate,

$$\lim_{n\to \infty }\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+\cdots+\frac{1}{a_na_{n-1}}$$

First I calculated few terms $a_1=1$, $a_2=5$, $a_3=9,a_4=13$ etc. So

$$\lim_{n\to \infty }\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+\cdots+\frac{1}{a_na_{n-1}}=\lim_{n\to \infty }\frac{1}{5}+\frac{1}{5\times9}+\cdots+\frac{1}{a_na_{n-1}}$$

Now I got stuck. How to proceed further? Should I calculate the sum ? Please help.

Answer

HINT:

$$\dfrac4{a_ma_{m-1}}=\dfrac{a_m-a_{m-1}}{a_ma_{m-1}}=?$$

$a_m=1+4\cdot(m-1)=?$

Do you recognize the Telescoping series?