# Recent questions tagged kenneth-rosen 1
Solve the recurrence relation for the number of rounds in the tournament described in question $14.$
2
How many rounds are in the elimination tournament described in question $14$ when there are $32$ teams?
3
Suppose that there are $n = 2^{k}$ teams in an elimination tournament, where there are $\frac{n}{2}$ games in the first round, with the $\frac{n}{2} = 2^{k-1}$ winners playing in the second round, and so on. Develop a recurrence relation for the number of rounds in the tournament.
4
Give a big-O estimate for the function $f$ given below if $f$ is an increasing function. $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$
1 vote
5
Find $f (n)$ when $n = 3k,$ where $f$ satisfies the recurrence relation $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$
6
Give a big-O estimate for the function $f$ in question $10$ if $f$ is an increasing function.
7
Find $f (n)$ when $n = 2^{k},$ where $f$ satisfies the recurrence relation $f (n) = f (n/2) + 1 \:\text{with}\: f (1) = 1.$
8
Suppose that $f (n) = f (n/5) + 3n^{2}$ when $n$ is a positive integer divisible by $5, \:\text{and}\: f (1) = 4.$ Find $f (5)$ $f (125)$ $f (3125)$
9
Suppose that $f (n) = 2f (n/2) + 3$ when $n$ is an even positive integer, and $f (1) = 5.$ Find $f (2)$ $f (8)$ $f (64)$ $(1024)$
10
Suppose that $f (n) = f (n/3) + 1$ when $n$ is a positive integer divisible by $3,$ and $f (1) = 1.$ Find $f (3)$ $f (27)$ $f (729)$
11
How many operations are needed to multiply two $32 \times 32$ matrices using the algorithm referred to in Example $5?$
12
Determine a value for the constant C in Example $4$ and use it to estimate the number of bit operations needed to multiply two $64$-bit integers using the fast multiplication algorithm.
13
Express the fast multiplication algorithm in pseudocode.
14
Multiply $(1110)_{2} \:\text{and}\: (1010)_{2}$ using the fast multiplication algorithm.
15
How many comparisons are needed to locate the maximum and minimum elements in a sequence with $128$ elements using the algorithm in Example $2$?
16
How many comparisons are needed for a binary search in a set of $64$ elements?
1 vote
17
Prove Theorem $6:$Suppose that $\{a_{n}\}$ satisfies the liner nonhomogeneous recurrence relation $a_{n} = c_{1}a_{n-1} + c_{2}a_{n-2} + \dots + c_{k}a_{n-k} + F(n),$ where $c_{1}.c_{2},\dots,c_{k}$ ... is $m,$ there is a particular solution of the form $n^{m}(p_{t}n^{t} + p_{t-1}n^{t-1} + \dots + p_{1}n + p_{0})s^{n}.$
18
Prove Theorem $4:$ Let $c_{1},c_{2},\dots,c_{k}$ be real numbers. Suppose that the characteristic equation $r^{k}-c_{1}r^{k-1}-\dots c_{k} = 0$ has $t$ distinct roots $r_{1},r_{2},\dots,r_{t}$ with multiplicities $m_{1},m_{2},\dots,m_{t},$ ... $\alpha_{i,j}$ are constants for $1 \leq i \leq t\:\text{and}\: 0 \leq j \leq m_{i} - 1.$
19
Solve the recurrence relation $T (n) = nT^{2}(n/2)$ with initial condition $T (1) = 6$ when $n = 2^{k}$ for some integer $k.$ [Hint: Let $n = 2^{k}$ and then make the substitution $a_{k} = \log T (2^{k})$ to obtain a linear nonhomogeneous recurrence relation.]
20
It can be shown that Cn, the average number of comparisons made by the quick sort algorithm (described in preamble to question $50$ in exercise $5.4),$ when sorting $n$ ... $48$ to solve the recurrence relation in part $(A)$ to find an explicit formula for $C_{n}.$
21
Use question $48$ to solve the recurrence relation $(n + 1)a_{n} = (n + 3)a_{n-1} + n, \:\text{for}\: n \geq 1, \:\text{with}\: a_{0} = 1$
22
Some linear recurrence relations that do not have constant coefficients can be systematically solved. This is the case for recurrence relations of the form $f (n)a_{n} = g(n)a_{n-1} + h(n).$ Exercises $48-50$ ... relation to obtain $a_{n} = \dfrac{C +\displaystyle{} \sum_{i = 1}^{n}Q(i)h(i)}{g(n + 1)Q(n + 1)}$
23
A new employee at an exciting new software company starts with a salary of $\$50,000$and is promised that at the end of each year her salary will be double her salary of the previous year, with an extra increment of$\$10,000$ for each year she has been with the ... $n^{\text{th}}$ year of employment. Solve this recurrence relation to find her salary for her $n^{\text{th}}$ year of employment.
24
Suppose that there are two goats on an island initially.The number of goats on the island doubles every year by natural reproduction, and some goats are either added or removed each year. Construct a recurrence relation for the number of goats on the island at the start of the ... $n^{\text{th}}$ year for each $n \geq 3.$
25
Suppose that each pair of a genetically engineered species of rabbits left on an island produces two new pairs of rabbits at the age of $1$ month and six new pairs of rabbits at the age of $2$ months and every month afterward. None of the rabbits ever die or leave ... relation in $(A)$ determine the number of pairs of rabbits on the island $n$ months after one pair is left on the island.
26
(Linear algebra required ) Let $A_{n}$ be the $n \times n$ matrix with $2s$ on its main diagonal, $1s$ in all positions next to a diagonal element, and $0s$ everywhere else. Find a recurrence relation for $d_{n},$ the determinant of $A_{n}.$ Solve this recurrence relation to find a formula for $d_{n}.$
27
Express the solution of the linear nonhomogenous recurrence relation $a_{n} = a_{n-1} + a_{n-2} + 1\:\text{for}\: n \geq 2 \:\text{where}\: a_{0} = 0\:\text{and}\: a_{1} = 1$ in terms of the Fibonacci numbers. [Hint: Let $b_{n} = a_{n + 1}$ and apply question $42$ to the sequence $b_{n}.]$
Show that if $a_{n} = a_{n-1} + a_{n-2}, a_{0} = s\:\text{and}\: a_{1} = t,$ where $s$ and $t$ are constants, then $a_{n} = sf_{n-1} + tf_{n}$ for all positive integers $n.$
Use the formula found in Example $4$ for $f_{n},$ the $n^{\text{th}}$ Fibonacci number, to show that fn is the integer closest to $\dfrac{1}{\sqrt{5}}\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n}$ Determine for which $n\: f_{n}$ ... $n\: f_{n}$ is less than $\dfrac{1}{\sqrt{5}}\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n}.$
Solve the simultaneous recurrence relations $a_{n} = 3a_{n-1} + 2b_{n-1}$ $b_{n} = a_{n-1} + 2b_{n-1}$ with $a_{0} = 1 \: \text{and}\: b_{0} = 2.$