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Time Complexity of Loop with Powers
• Difficulty Level : Medium
• Last Updated : 19 Oct, 2020

What is the time complexity of the below function?

## C

 `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)

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