I have seen this formula from Ramanujan
$$\sum_n \frac{a^{n+1}-b^{n+1}}{a-b}\frac{c^{n+1}-d^{n+1}}{c-d}T^n = \\\frac{1-abcdT^2}{(1-acT)(1-adT)(1-bcT)(1-bdT)} \tag{1}$$
I know how to prove it via geometric series but I don't find this proof very intuitive:
$$ \frac{1}{(a-b)(c-d)}\left(\frac{ac}{1-acT} - \frac{ad}{1-adT} - \frac{bc}{1-bcT} + \frac{bd}{1-bdT} \right) = \\ \frac{1-abcdT^2}{(1-acT)(1-adT)(1-bcT)(1-bdT)} \tag{2}$$
Does anyone have an idea how Ramanujan got this formula? Or, more generally, does anyone have an idea how to prove it intuitively?