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#3: Post edited by user avatar leovt‭ · 2021-08-14T13:38:32Z (over 3 years ago)
Tried to explain more clearly after feedback in the comment.
  • 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 $k=0$ and not $k+1=1$.
  • 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.
  • 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.
#2: Post edited by user avatar leovt‭ · 2021-08-11T16:55:28Z (over 3 years ago)
  • 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 $k=0$ and not $k+1=1$.
  • 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
  • $$w_{k+1} - w_0 = r^k (w_1-w_0)$$ as a proper proof by induction.
  • 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 $k=0$ and not $k+1=1$.
  • 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.
#1: Initial revision by user avatar leovt‭ · 2021-08-11T16:54:38Z (over 3 years ago)
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 $k=0$ and not $k+1=1$.

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 
$$w_{k+1} - w_0 = r^k (w_1-w_0)$$ as a proper proof by induction.