When solving the Gambler’s Ruin recurrence
[ P_i = pP_{i+1} + qP_{i-1}, ]
a common mistake is to assume
[ P_i = i^r. ]
Substituting this gives
[ i^r = p(i+1)^r + q(i-1)^r, ]
which is difficult to simplify and does not lead to an easy solution.
Instead, for linear recurrences with constant coefficients, we use the trial solution
[ P_i = r^i. ]
This is useful because shifting the index only multiplies the expression by a constant:
[ P_{i+1} = r^{i+1}, \qquad P_{i-1} = r^{i-1}. ]
Substituting into the recurrence yields
[ r^i = pr^{i+1} + qr^{i-1}. ]
Dividing through by r^{i-1} gives
[ r = pr^2 + q, ]
or equivalently,
[ pr^2 – r + q = 0. ]
This quadratic equation, called the characteristic equation, can then be solved to obtain the general solution of the recurrence.
The key idea is that exponential forms such as r^i preserve their structure when the index changes, making them ideal candidates for solving linear difference equations.