$1.$ Consider the following code and determine how many times "Grape" will be printed to the console. For this problem, also state what the value of f$($n$)$ is$.$ Your solution should be written as O$($some function$)=$ true running time, where you need to specify the function for g$($n$)$ and the true running time for f$($n$).$

for $($int i $=0$; i $<25*$ n $+5$; i $++)\{$

System $.$ out $.$ println $("$ Grape $")$ ;

$\}$

for $($int i $=0$; i $<30*$ n ; i $++)\{$

System $.$ out $.$ println $("$ Grape $")$ ;

$\}$

$2.$ Consider the equation f$($n$)=123$n and g$($n$)=$ n$2.$Suppose we randomly picked c $=15.$ What is the smallest n$0($where n$0>0)$ such that $$n $>=$ n$0,0<=$ f$($n$)<=$ c$$g$($n$).$

State your solution clearly.

$3.$ What does n represent in the definition for O$()$ notation?

$4.$ Consider the following code and determine how many times "Grape" will

be printed to the console. The solution should be stated in O$()$ notation.

for $($int i $=0$; i $<$ n ; i $++)\{$

for $($int k $=0$; k $<$ n $*$ n ; k $++)\{$

System $.$ out $.$ println $("$ Grape $")$ ;

$\}$

$\}$

$5.$ Demonstrate the output for the following code and state the asymptotic complexity.

Assume that n $=6$ for this problem.

for $($int i $=1$; i $<$ n ; i $++)\{$

for $($int k $=0$; k $<$ i ; k $++)\{$

System $.$ out $.$ print $("*")$ ;

$\}$

System $.$ out $.$ println $()$ ;f

$\}$

$6.$ Consider the following code and determine how many times "Grape" will be printed to the console. The solution should be stated in O$()$ notation

for $($int i $=0$; i $<$ n ; i $++)\{$

for $($int k $=0$; k $<$ n ; k $++)\{$

for $($ int m $=0$; m $<$ i ; m $++)\{$

System $.$ out $.$ println $("$ Grape $")$ ;

$\}$

$\}$

$\}$

$7.$ Consider the following code and determine how many times "Grape" will be printed to the console. The solution should be stated in O$()$ notation

for $($int i $=0$; i $<$ n ; i $++)\{$

for $($int k $=0$; k $<$ i ; k $++)\{$

for $($ int m $=0$; m $<$ k ; m $++)\{$

System $.$ out $.$ println $("$ Grape $")$ ;

$\}$

$\}$

$\}$

$8.$ Consider the following code and determine how many times "Grape" will be printed to the console. The solution should be stated in O$()$ notation

for $($int i $=0$; i $<$ n ; i $++)\{$

for $($int k $=0$; k $<100$; k $++)\{$

for $($ int m $=0$; m $<$ n ; m $++)\{$

System $.$ out $.$ println $("$ Grape $")$ ;

$\}$

$\}$

$\}$

$9.$ Consider the following code and determine how many times "Grape" will be printed to the console. The solution should be stated in O$()$ notation

int i $=$ n ;

while $($ i $>=1)\{$

for $($ int k $=0$; k $<$ n ; k $++)\{$

System $.$ out $.$ println $("$ Grape $")$ ;

$\}$

i $=($int$)$ i $/3$;

$\}$