Several real numbers (not necessarily are different) are chosen. The sum of these numbers is 10. Is it possible that the sum of the squares of these numbers is less than one-millionth? Justify your answer.
Any hints or help would be greatly appreciated. Thanks!
Since you don't know how many numbers are chosen, you can't just pick a specific number. But Laurent showed you that you can choose all equal numbers, which are is possible by the hypothesis, then compute the sum of their squares. You should follow Laurent's hint.
Square each number, each of them is , so its square is .
Then add up all of these square times. See what you get, and check if it is possible that the sum you just computed is less than one-millionth.
And the point of looking at all numbers equal is that you can show that the more they differ, the larger the sum of squares will be. The case with all numbers equal is the smallest sum of squares. If that sum can't be "less than one-millionth" then the other sums certainly can't be.
Ok I kinda get what you're saying here. By using your equation above I can plug in 100,000,000 for n and get exactly one-millionth. The problem I'm having with this is how the sum of these numbers can even equal 10. This is why I tried to get the sum to 10 in my previous post but couldn't.
To everyone that's posted, I will give thanks when I figure this out but I don't see the picture everyone is trying to paint here.