Don't understand the explanation of "sum of the factors In 5!"
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amusingminnow M0 M1 M2 M3 M4 M5last edited by amusingminnow Jul 3, 2021, 2:28 PM Jul 3, 2021, 2:15 PM
Why does 2^0+2^1+2^2+2^3=2^4-1 ?
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@amusingminnow Great question! This is a pattern that works for all powers of two, actually.
Let's add them one at a time and see if there's anything you notice:
What's special about all of the answers is that they're all less than the next power of !Knowing this is super helpful, it saves us from having to do a ton of addition. For example, we can use this shortcut when doing , because we know that this is equal to the next power of which is , minus !
So .
If you really want to get into the nitty gritty of why this works, you have to factor. Let's use .
Remember difference of squares? We can use that here!
And since is a difference of two squares also, we can keep going.
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Well, is just , so the expression is equal to . Expand , and you'll get .So,
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You can use factoring to prove this for all of the powers of