Problem 1

Solve each homogeneous linear recurrence relations.

1. ${\displaystyle f(n+1)=5f(n)-6f(n-1)}$
2. ${\displaystyle f(n+1)=4f(n)}$
3. ${\displaystyle f(n+1)=6f(n)+7f(n-1)+6f(n-2)}$

Answer

1. We express the relation in matrix form.

   ${\displaystyle {\begin{pmatrix}5&-6\\1&0\end{pmatrix}}{\begin{pmatrix}f(n)\\f(n-1)\end{pmatrix}}={\begin{pmatrix}f(n+1)\\f(n)\end{pmatrix}}}$

   The characteristic equation of the matrix

   ${\displaystyle {\begin{vmatrix}5-\lambda &-6\\1&-\lambda \end{vmatrix}}=\lambda ^{2}-5\lambda +6}$

   has roots of ${\displaystyle 2}$ and ${\displaystyle 3}$. Any function of the form ${\displaystyle f(n)=c_{1}2^{n}+c_{2}3^{n}}$ satisfies the recurrence.

2. This is like the prior part, but simpler. The matrix expression of the relation is

   ${\displaystyle {\begin{pmatrix}4\end{pmatrix}}{\begin{pmatrix}f(n)\end{pmatrix}}={\begin{pmatrix}f(n+1)\end{pmatrix}}}$

   and the characteristic equation of the matrix

   ${\displaystyle {\begin{vmatrix}4-\lambda \end{vmatrix}}=4-\lambda }$

   has the single root ${\displaystyle 4}$. Any function of the form ${\displaystyle f(n)=c4^{n}}$ satisfies this recurrence.

3. In matrix form the relation

   ${\displaystyle {\begin{pmatrix}6&7&6\\1&0&0\\0&1&0\end{pmatrix}}{\begin{pmatrix}f(n)\\f(n-1)\\f(n-2)\end{pmatrix}}={\begin{pmatrix}f(n+1)\\f(n)\\f(n-1)\end{pmatrix}}}$

   gives this characteristic equation.

   ${\displaystyle {\begin{vmatrix}6-\lambda &7&6\\1&-\lambda &0\\0&1&-\lambda \end{vmatrix}}=-\lambda ^{3}+6\lambda ^{2}+7\lambda +6}$

Problem 2

Give a formula for the relations of the prior exercise, with these initial conditions.

1. ${\displaystyle f(0)=1}$, ${\displaystyle f(1)=1}$
2. ${\displaystyle f(0)=0}$, ${\displaystyle f(1)=1}$
3. ${\displaystyle f(0)=1}$, ${\displaystyle f(1)=1}$, ${\displaystyle f(2)=3}$.

Problem 3

Check that the isomorphism given between ${\displaystyle S}$ and ${\displaystyle \mathbb {R} ^{k}}$ is a linear map. It is argued above that this map is one-to-one. What is its inverse?

Problem 4

Show that the characteristic equation of the matrix is as stated, that is, is the polynomial associated with the relation. (Hint: expanding down the final column, and using induction will work.)

Problem 5

Given a homogeneous linear recurrence relation ${\displaystyle f(n+1)=a_{n}f(n)+\dots +a_{n-k}f(n-k)}$, let ${\displaystyle r_{1}}$, ..., ${\displaystyle r_{k}}$ be the roots of the associated polynomial.

1. Prove that each function ${\displaystyle f_{r_{i}}(n)=r_{k}^{n}}$ satisfies the recurrence (without initial conditions).
2. Prove that no ${\displaystyle r_{i}}$ is ${\displaystyle 0}$.
3. Prove that the set ${\displaystyle \{f_{r_{1}},\dots ,f_{r_{k}}\}}$ is linearly independent.

Problem 6

(This refers to the value ${\displaystyle T(64)=18,446,744,073,709,551,615}$ given in the computer code below.) Transferring one disk per second, how many years would it take the priests at the Tower of Hanoi to finish the job?