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# In $w_{k + 1} - w_k = (\frac{1 - p}{p})^{exponent}(w_1 - w_0)$, why isn't exponent $k + 1$?

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Please see the $r^k$ underlined in red, which is $(\frac{1 - p}{p})^k$ as defined by the green underlines.

1. How do you deduce that the exponent must be $k$? Why isn't the exponent $k + 1$?

2. Is this question related to the Fence Post Error? Have I committed it? Tsitsiklis, Introduction to Probability (2008 2e), p 63.

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If you set $k=0$ the equation becomes $$w_{0+1} - w_0 = r^{\textrm{exponent}}(w_1-w_0)$$ For this equation to hold (in general) the exponent must be zero and not one. Thus an exponent of $k+1$ must be wrong.

I think the problem is not a fence-post problem but the proper base case for the induction. Try to write down the step of going from $$w_{k+1} - w_k = r (w_k-w_{k-1})$$ to $$w_{k+1} - w_k = r^k (w_1-w_0)$$ as a proper proof by induction.

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