We take number $k,l$ from two sets - what is expected value of number $l$?

We take number $k,l$ from two sets - what is expected value of number $l$?

We take number $k$ from set $\left\{1,2,...,n \right\} $ and then we take
number $l$ from set $\left\{1,2,...,k \right\} $. Let $E_n$ be expected
value of number $l$. Compute:
$E_2$
$E_3$
$E_4$
$E_5$
I have computed it and I received:
$E_2 = \frac{5}{4}$
$E_3 = \frac{3}{2}$
$E_4 = \frac{7}{4}$
$E_5 = 2$
It was quite easy but very long. Maybe exist faster way to compute it?
As we can see:
$E_3 = \frac{3}{2} = \frac{6}{5}E_2 \\ E_4 = \frac{7}{4} = \frac{7}{5} E_2
\\ E_5 = 2 = \frac{8}{5}E_2$
So I suppose that for all $n \ge 3$ we have $E_n = \frac{n+3}{5} E_2$ but
I have no idea how can I prove it exactly.