# Tagged Questions

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### Proof of a sequence with recursion

The problem asks to prove the following to be true. $$F^2_{n+1} - F_{n+1} F_n - F_n^2 = (-1)^n$$ Anyway, I've tried looking at this or similar proofs for going on an hour now, pretty much the only ...
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### Recursive formulae involving a linear operator

Given a basis $e_{1}$, $e_{2}$ in the plane, define the linear operator $F$ as $F(e_{1})=3e_{1}+e_{2}$ and $F(e_{2})=e_{2}$. Furthermore, define the sequence $u_{1},u_{2},\dots$ of vectors in the ...
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### How to find an explicit formula for $d _ k$

Consider sequence $d _ 1, d _2, d_3$ $$d_k= \frac {d_{k-1}} {k + 1}$$ for all integers $k \ge 2$ with the initial condition that $d_1 = 1$. Find an explicit formula for $d_k$ for the $k^{th}$ ...
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### Proof by induction for a recursive function $F(n) = F(n–1)+F(n–2)$

I'm having a lot of trouble with the following homework question: Consider the following recursive function: Base Case: $F(0) = 0,F(1) = 1$. Recursive Step: $F(n)=F(n−1)+F(n−2)$ for all ...
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