On 20:00 of this video (https://www.youtube.com/watch?v=PNrqCdslGi4&index=7&list=PL2SOU6wwxB0uwwH80KTQ6ht66KWxbzTIo) the professor explains about the Gambler's Ruin problem.
He goes from $$x^i = px^{i+1}+qx^{i-1}$$
to
$$px^2-x+q=0$$
I do not understand why the first equation has the i th indicator on it but the second doesn't. How can he goes from the first one with sequence indicator (i th number) into equation with no sequence indicator ?
Can anyone please show me very step by step algebra with detail explanation about why it is valid to goes from the first equation to second ?
I have no experience about difference equation and a little knowledge on differential equation.
you asked-" I do not understand why the first equation has the i th indicator on it but the second doesn't. How can he goes from the first one with sequence indicator (i th number) into equation with no sequence indicator ? "
The first equation is general and holds for any integer value of i.
the 2nd equation has no i in it because it is the specific case of the first equation when i=1, (and subtracting x from both sides of that equation).