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I wanted to double check my understanding and working out for a 3 nested for loop algorithm, and working out it's complexity. I've got the right answer, but how I've arrived at it I feel isn't exactly correct.

$\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}\sum_{k=0}^{n-1} 1\\$

$\sum_{i=0}^{n-1}\sum_{j=0}^{n-1} n\\$

$\sum_{i=0}^{n-1}\sum_{j=0}^{n-1} n*1\\$

$n\sum_{i=0}^{n-1}\sum_{j=0}^{n-1} 1\\$

$n\sum_{i=0}^{n-1}n\\$

$n^2\sum_{i=0}^{n-1}1\\$

$n^3\\$

Essentially, I move the $n$ outside the summations, based on the sum manipulation rules. However, I feel this is incorrect given it should only be constants that are moved outside the summation.

If that's the case, I'm not too sure how to arrive at the $n^3$ answer, if we can't "move" the $n$ outside. Any help would be much appreciated.

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  • $\begingroup$ It is valid to move the $n$ outside of the summations because all of the summations are finite and have indices that are not dependent on $n$. $\endgroup$ – Peter Foreman Apr 16 at 22:44
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It is correct. You can move any multiplier that doesn't depend on index of summation, and $n$ doesn't depend on $i$, $j$, $k$.

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  • $\begingroup$ That makes perfect sense, thank you for that mihaild. I checked over the summation rules, and I didn't realise that it mentions the index. $\endgroup$ – WhatToDisplay Apr 16 at 22:49

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