How do I find this function the growth rate based on the big O notation?

Question

How do I find this function the growth rate based on the big O notation?

for(i=1; i*i<n; i=i+1)
    for(j=1; j<=i; j=j+1)

Answer 1

See the following

when i=1 , innerloop runs 1 time

outloop i=2 , innerloop runs 2 times

outloop i=3 , innerloop runs 3 times

outloop i=4 , innerloop runs 4 times

....

outloop i=sqrt(n)-1 , innerloop runs sqrt(n)-1 times, the loop terminates here

Lest assume that n is a perfect square

==> f(n) = 1 + 2 + 3 + 4 + ... + sqrt(n) - 1 = (sqrt(n) - 1 )/2 x ( sqrt(n) - 1 + 1)

==> f(n) = ( (sqrt(n)^2)/2 - sqrt(n)/2 

==> O( sqrt(n)^2)/2 - sqrt(n)/2 ) = O( n - sqrt(n) )

==> O( n - sqrt(n) )
==> O( n )

So it approximately takes O(n)