Determine the positive number that must be added or subtracted to the expression to complete the square. x ^ 2 - 2x.
1 answer:
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1. The terms of a sequence are denoted by 
2.
3. so it is clear that the first columns add each time by one, and the second column add by 2, then by 4, by 6, by 8 and so on.
4. consider only the second column and how we get the terms, which we will call
:
5.
So
![u_{n}=(n+1)(1+2{1+2+3+....(n-1)}) =(n+1)(1+2 [(n-1)n/2]) = (n+1)(1+(n-1)n) =(n+1)( n^{2}-n+1 ) ](https://tex.z-dn.net/?f=u_%7Bn%7D%3D%28n%2B1%29%281%2B2%7B1%2B2%2B3%2B....%28n-1%29%7D%29%0A%20%20%20%20%20%20%20%20%0A%20%20%20%20%20%20%20%20%20%3D%28n%2B1%29%281%2B2%20%5B%28n-1%29n%2F2%5D%29%0A%0A%20%20%20%20%20%20%20%20%20%3D%20%28n%2B1%29%281%2B%28n-1%29n%29%0A%20%20%20%20%20%20%20%0A%20%20%20%20%20%20%20%20%20%3D%28n%2B1%29%28%20n%5E%7B2%7D-n%2B1%20%29%0A%20%20%20%20%20%20%20%20%20)
6. We can check:
7. Remark: Gauss addition formula: 1+2+3+....+n=n(n+1)/2
2.
3. so it is clear that the first columns add each time by one, and the second column add by 2, then by 4, by 6, by 8 and so on.
4. consider only the second column and how we get the terms, which we will call
5.
So
6. We can check:
7. Remark: Gauss addition formula: 1+2+3+....+n=n(n+1)/2