discrete mathematics - Using Direct Proof. $1+2+3+ldots+n = frac{n(n + 1)}{2}$

Thursday 27 February 2014

I need help proving this statement. Any help would be great!

Answer

Here is an approach.

$$ s_n =1+2+3+\dots+(n-1)+n \\
s_n =n+(n-1)+(n-2)+\dots+1 . $$

Adding the above gives

$$2s_n = (1+n)+(2+(n-1))+(3+(n-2))+\dots+(1+n) $$

$$ =(1+n)+(1+n)+\dots+(1+n) $$

The above is nothing but adding $(1+n)$ n times and the result follows

$$ \implies s_n = \frac{n(n+1)}{2}. $$

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