$1$ For each $of$ the following codes:

i Find the exact mathematical complexity

$ii$ find a cost model which $is$ the most frequently operation.

iii Find the Telda approximation

$iv$ Finally, find the order $of$ growth

$as$ a function $ofN$:

$a.$ int sum $=0$;

for int $j=N$; $j>0$; $j\frac{?}{?}=2$

forint $i=0$; $i<j$; $i++$

sum$++$;

$b.$ int sum $=0$;

for int $i=1$; $i<N$; $i*=2$

for int $j=0$; $j<i$; $j++$

sum$++$;

$c.$ int sum $=0$;

for int $i=1$; $i<N$; $i*=2$

for int $j=0$; $j<N$; $j++$

sum$++$;

$d.$ int sum $=0$;

for int $i=1$; $i<N$; $i+=3$

sum$++$;