Answer
a) 4
b) 61
c) 509
d) 8189
Work Step by Step
We are given
$$f(n)=2f(n/2)+3$$
and $$f(1)=5$$
a) Apply the recurrence relation when $n=2$
$f(2)=2f(1)+ 3$
$=1+3$
$=4$
b) Apply the recurrence relation when $n=8$
First we find $f(4)=2f(2)+3$
$=2\times13+3$
$=26+3$
$=29$
Then find $f(8)=2f(4)+3$
$=2\times29+3$
$=58 + 3$
$=61$
c) Apply the recurrence relation with $n=64$
First we find $f(16)=2f(8)+3$
$=2\times61+3$
$=125$
$f(32)=2f(16)+3$
$=2\times125+3$
$=253$
Then find $f(64)=2f(32)+3$
$=2\times253+3$
$=509$
d) Apply the recurrence relation with $n=1024$
First we find $f(128)=2f(64)+3$
$=2\times509+3$
$=1021$
$f(256)=2f(128)+3$
$=2\times1021+3$
$=2045$
$f(512)=2f(256)+3$
$=2\times2045+3$
$=4093$
Then find $f(1024)=2f(512)+3$
$=2\times4093+3$
$=8189$
