Time complexity for dependant nested for loop?

Question

Can you explain me how to find time complexity for this?

sum=0;
for(k=1;k<=n;k*=2)
  for(j=1;j<=k;j++)
     sum++;

So, i know the out

Answer 1

Just see how many times the inner loop runs:

1 + 2 + 4 + 8 + 16 +...+ n

Note that if n = 32, then this sum = 31 + 32. ~ 2n.
This is because the sum of all the terms except the last term is almost equal to the last term.

Hence the overall complexity = O(n).

EDIT:

The geometric series sum (http://www.mathsisfun.com/algebra/sequences-sums-geometric.html) is of the order of:

(2^(logn) - 1)/(2-1) = n-1.