Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Suppose that $f (n) = 2 f (n / 2) + 3$ when $n$ is an even positive integer, and $f (1) = 5$ . Find
a) $f (2)$
b) $f (8)$ .
c) $f (64)$ .
d) $f (1024)$ .

The answers are given below:

(a) $f (2) = 13$

(b) $f (8) = 61$

(d) $f (1024) = 8189$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Recurrence Relation definition

A recurrence relation is an equation that recursively defines a sequence where the next term is a function of the previous terms: $f(n) = a f(n / b) + c$

Step 2: Apply Recurrence Relation

Given:$f(n) = 2f(n/2) + 3\,{\rm{and}}\,f(1) = 5$

(a) Repeatedly apply the recurrence relation until we obtain $n = 3$ :

$\begin{aligned}f(2) &= 2f(1) + 3\\ &= 2(5) + 3\\ &= 13\end{aligned}$

(b) Repeatedly apply the recurrence relation until we obtain $n = 8$ :

$\begin{aligned} f(4) &= 2f(2) + 3\\ &= 2(13) + 3\\ &= 29\\f(8) &= 2f(4) + 3\\ &= 2(29) + 3\\& = 61\end{aligned}$

$\begin{aligned}f(16) &= 2f(8) + 3\\f(16) &= 2(61) + 3\\ &= 125\\f(32) &= 2f(16) + 3\\ &= 2(125) + 3\\ &= 253\\f(64) &= 2f(32) + 3\\ &= 2(253) + 3\\ &= 509\end{aligned}$

(d) Repeatedly apply the recurrence relation until we obtain$n = 1024$:

$\begin{aligned}f(128) &= 2f(64) + 3\\ &= 2(509) + 3\\ &= 1021\\f(256) &= 2f(128) + 3\\& = 2(1021) + 3\\ &= 2045\\f(512) &= 2f(256) + 3\\ &= 2(2045) + 3\\ &= 4093\\f(1024) &= 2f(512) + 3\\ &= 2(4093) + 3\\ &= 8189\end{aligned}$

Find Study Materials for

Create Study Materials

Select your language

Discrete Mathematics and its Applications

Answers without the blur.

Short Answer

Suppose that $f (n) = 2 f (n / 2) + 3$ when $n$ is an even positive integer, and $f (1) = 5$ . Find
a) $f (2)$
b) $f (8)$ .
c) $f (64)$ .
d) $f (1024)$ .

Step by Step Solution

Step 1: Recurrence Relation definition

Step 2: Apply Recurrence Relation

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Discrete Mathematics and its Applications

Answers without the blur.

Short Answer

Suppose that f(n)=2f(n/2)+3 when n is an even positive integer, and f(1)=5. Finda)f(2)b)f(8).c)f(64).d)f(1024).

Step by Step Solution

Step 1: Recurrence Relation definition

Step 2: Apply Recurrence Relation

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Suppose that $f (n) = 2 f (n / 2) + 3$ when $n$ is an even positive integer, and $f (1) = 5$ . Find
a) $f (2)$
b) $f (8)$ .
c) $f (64)$ .
d) $f (1024)$ .