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**Prove that summation of n(n+1)/2 is true for all integers. Why is my proof not valid?**

Could someone explain to me how this is not a valid proof of the summation of "i" from i=1 to n:

n(n+1)/2

Show for base cases:

n=1: 1(1+1)/2=1

n=2: 2(2+1)/2=3

n=3: 3(3+1)/2=6

...

inductive hypothesis: 1+2+...+(n-1)=n(n-1)/2

((n-1)*n)/2 + n =

((n-1)*n)/2 + 2*n/2 =

((n-1)*n + 2*n)/2 =

((n-1 + 2)*n)/2 =

((n+1)*n)/2 =

(n*(n+1))/2.

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