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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.

user3270418
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  • Divide the equation by $x^{j-1}$... – Fabian Nov 26 '16 at 06:57
  • Lookup the characteristic polynomial for homogeneous linear recurrences. – dxiv Nov 26 '16 at 06:58
  • homogeneous is a new word for me. Should I also finish my differential equation class before to get deeper insight in this topic? – user3270418 Nov 26 '16 at 07:01
  • You don't need calculus for linear recurrences. Just notice that $p r^2 - r + q = 0 \iff r = p r^2+q$ then multiply by $r^{i-1}$ to verify that it satisfies the recurrence. – dxiv Nov 26 '16 at 07:10
  • x^i dosen't have a specific number why in the second equation there is no x^i??? please give me step by step and detail explanation from first equation to the second. I'm newbie – user3270418 Nov 26 '16 at 07:12
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    Because if you assume the solutions are of the form $x^i$ then you can divide the first equation by $x^{i-1}$ and you get the second one. See the previous link, maybe also this or this. – dxiv Nov 26 '16 at 07:20

1 Answers1

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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).

steve newman
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