math - Time complexity of a function with time 1 + 8 + 27 + 64 + ... + sqrt(n)^3? -

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i have been told that

1 + 8 + 27 + 64 + ... + (√n)3 = θ(n2)

why case?

to make sure understand you're saying, curious in why sum

13 + 23 + 33 + ... + (√n)3 = θ(n2)

one way a formula sum of first m cubes. equal to

(m(m + 1) / 2)2

so let's plug in m = √n, gives

13 + 23 + 33 + ... + (√n)3

= ((√n)((√n) + 1) / 2)2

= ((n + √n) / 2)2

= (n2 + 2n√n + n) / 4

this final expression gives exact value of sum of first √n perfect cubes. note expression θ(n2), because n2 dominant term.

hope helps!


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