Question

2+4+6........+100-(1+3+5+...+99)=

Answer 1

2+4+6+....+100 -
1 -3 -5 -......- 99
______________
1+1+1+.........+1

2+4+6........+100-(1+3+5+...+99)= 1*50=50

Answer 2

$\displaystyle 2+4+6+...+100-(1+3+5+...+99) \\ \\ 2+4+6+...+100=2(1+2+3+...+50)=\not2 \cdot \frac{50(50+1)}{\not2} = \\ \\ =50 \cdot 51=2550 \\ \\ 2+4+6+...+100=2550$

$\displaystyle 1+3+5+...+99 \\ \\ 99=1+(n-1) \cdot 2 \Rightarrow 99=1+2n-2 \Rightarrow 2n=99-1+2 \Rightarrow \\ \\ \Rightarrow 2n=100 \Rightarrow n= \frac{100}{2} \Rightarrow n=50\\ \\ S_{50}= \frac{2 \cdot 1+(50-1) \cdot 2}{2} \cdot 50 \\ \\ S_{50}=(2+49 \cdot2)\cdot 25 \\ \\ S_{50}=(2+98) \cdot 25 \\ \\ S_{50}=100 \cdot 25\\ \\ S_{50}=2500 \\ \\1+3+5+...+99=2500 \\ \\ 2+4+6+...+100-(1+3+5+...+99)=2550-2500=\boxed{50}$