Question

Create a recurrence equation( T(n) ) and solve to get O(n)

A(int n)

if (n==1)

time++;

else{

for i=1 to i->n

time++;

A(n-1)

}

Answer 1

A(int n)
if (n==1)
    time++;
else{
    for i=1 to i->n
        time++;
    A(n-1)
}

In each recursive call, a work(for loop) of n iterations is done. and a recursive call of size n-1 is made.
so, recurrence relation is T(n) = T(n-1) + O(n)

T(n)
= T(n-1) + n
= T(n-2) + n-1 + n
= T(n-3) + n-2 + n-1 + n
= T(1) + 2 + ... + n-2 + n-1 + n
= 1 + 2 + ... + n-2 + n-1 + n
= n(n+1)/2
so, T(n) = O(n^2)

so, time complexity is O(n^2)