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I made a summation based off of Mathologers video from 3 months ago called:

‘What’s hiding beneath? Animating a mathematical gem’ (7:30) for those who want to see.

I tried my best to make it correct but I’m not sure if it is or if it is the most simplified way to do it.

Thanks!
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I’m not sure what the question is, but work on summation notation.

The first line should be:

The sum from k=0 to n-1 of (3+k)^n is approximately (3+n)^n
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I don’t know what the video is getting at, since it has “= ?^4 “, not approximately. Also I don’t know if they wanted you to keep going.
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When you say “Sum ____”, the description you include of each term is *each* term. For example:  
“Sum each number” S x = 1 + … + n  
“Sum each square” S x^2 = 1^2 + … + n^2

If it helps you remember, the capital S is just the initial of Sum.

Other than that, yes, you have described the sum correctly. The point is to describe it *instead of* fully writing it down.

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