# Time Complexity of Loop with Powers

What is the time complexity of the below function?

 `void` `fun(``int` `n, ``int` `k)` `{` `    ``for` `(``int` `i = 1; i <= n; i++) ` `    ``{` `        ``int` `p = ``pow``(i, k);` `        ``for` `(``int` `j = 1; j <= p; j++) ` `        ``{` `            ``// Some O(1) work` `        ``}` `    ``}` `}`

Time complexity of above function can be written as 1k + 2k + 3k + … n1k.

Let us try few examples:

```k=1
Sum = 1 + 2 + 3 ... n
= n(n+1)/2
= n2/2 + n/2

k=2
Sum = 12 + 22 + 32 + ... n12.
= n(n+1)(2n+1)/6
= n3/3 + n2/2 + n/6

k=3
Sum = 13 + 23 + 33 + ... n13.
= n2(n+1)2/4
= n4/4 + n3/2 + n2/4
```

In general, asymptotic value can be written as (nk+1)/(k+1) + Θ(nk)
If  n>=k then the time complexity will be considered in O((nk+1)/(k+1)) and if n<k, then the time complexity will be considered as  in the O(nk)

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.

Article Tags :