[tex]\displaystyle a).2+4+6+...+100=2(1+2+3+...+50)=\not 2 \cdot \frac{50(50+1)}{\not 2} = \\ \\ =50 \cdot 51=2550 \not = p.p. \\ \\ b).1+3+5+...+199 \\ \\ 199=1+(n-1) \cdot 2 \Rightarrow 199=1+2n-2 \Rightarrow 2n=199-1+2 \Rightarrow \\ \\ \Rightarrow 2n=200 \Rightarrow n= \frac{200}{2} \Rightarrow n=100 \\ \\ S_{100}= \frac{2 \cdot 1+(100-1) \cdot 2}{2} \cdot 100 \\ \\ S_{100}=(2+99 \cdot 2) \cdot 50 \\ \\ S_{100}=(2+198) \cdot 50 \\ \\ S_{100}=200 \cdot 50\\ \\ S_{100}=10000 \\ \\ S_{100}=100^2=p.p.[/tex]
