Fibonacci numbers

6.1 Why do we have to specify exactly two of the elements to begin with? Why not one or three?

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6.2 We have $n$ dollars to spend. Every day we either buy a candy for 1 dollar, or an icecream for 2 dollars. In how many ways can we spend the money?

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6.3 Prove that $F_{3n}$ is even.

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6.4 Prove that $F_{5n}$ is divisible by 5.

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6.5 Prove the following identities.

(a) $F_1+F_3+F_5+\cdots +F_{2n-1}=F_{2n}$.

(b) $F_0-F_1+F_2-F_3+\cdots -F_{2n-1}+F_{2n}=F_{2n-1}-1$.

(c) $F_0^2+F_1^2+F_2^2+\cdots +F_n^2=F_n\cdot F_{n+1}$.

(d) $F_{n-1}{F}_{n+1}-F_n^2=(-1{)}^n$.

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6.6 Mark the first entry (a 1) of any row of the Pascal triangle. Move one step East and one step Northeast, and mark the entry there. Repeat this until you get out of the triangle. Compute the sum of the entries you marked.

(a) What numbers do you get if you start from different rows? First “conjecture”, then prove your answer.

(b) Formulate this fact as an identity involving binomial coefficients.

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6.7 Cut a chessboard into 4 pieces as shown in Figure 10 and assemble a $5\times 13$ rectangle from them. Does this prove that $5\cdot 13=8^2$? Where are we cheating? What does this have to do with Fibonacci numbers?

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6.8 Prove Theorem 6.1 by induction on $n$.

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6.9 Define a sequence of integers $H_n$ by $H_0=1$, $H_1=3$, and $H_{n+1}=H_n+H_{n-1}$. Show that $H_n$ can be expressed in the form $a\cdot q_1^n+b\cdot q_2^n$ (where $q_1$ and $q_2$ are the same numbers as in the proof above), and find the values of $a$ and $b$.

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6.10 Define a sequence of integers $I_n$ by $I_0=0$, $I_1=1$, and $I_{n+1}=4I_n+I_{n-1}$. Find a formula for $I_n$.

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