[tex]\displaystyle a).1+3+5+...+99= \\ \\ =1+2+3+4+5+...+99-(2+4+6+...+98)= \\ \\ = \frac{99(99+1)}{2} -2(1+2+3+...+49)= \frac{99 \times 100}{2} -2 \times \frac{49(49+1)}{2} = \\ \\ = \frac{9900}{2} -2 \times \frac{49 \times 50}{2} =4950- \not2 \times \frac{2450}{\not2} =4950-2450=2500[/tex]
[tex]\displaystyle b).2+4+6+...+100=2(1+2+3+...+50)=2 \times \frac{50(50+1)}{2} = \\ \\ =2 \times \frac{50 \times 51}{2} =\not 2 \times \frac{2550}{\not2} =2550[/tex]
[tex]\displaystyle c).3+7+11+15+...+43 \\ 3=1 \times 4-1 \\ 7=2 \times 4-1 \\ 11=3 \times 4-1 \\ 15=4 \times 4-1 \\ . \\ . \\ . \\ 43=4 \times 11-1 \\ 3+7+11+15+...+43=4(1+2+3+4+...+11)-1 \times 11 \\ \\ 3+7+11+15+...+43= 4 \times \frac{11 \times 12}{2} -11 \\ \\ 3+7+11+15+...+43=4 \times \frac{132}{2} -11 \\ \\ 3+7+11+15+...+43=4 \times 66-11 \\ \\ 3+7+11+15+...+43=264-11 \\ \\ 3+7+11+15+...+43=253[/tex]
[tex]\displaystyle d).3+6+12+...+2001 =3(1+2+3+...+667)= \\ \\ =3 \times \frac{667(667+1)}{2} =3 \times \frac{667 \times 668}{2} =3 \times \frac{445556}{2} = \\ \\ =3 \times 222778=668334[/tex]
[tex]\displaystyle e).5+10+15+...+2000=5(1+2+3+...+400)= \\ \\ =5 \times \frac{400(400+1)}{2} =5 \times \frac{400 \times 401}{2} =5 \times \frac{160400}{2} = \\ \\ =5 \times 80200=401000[/tex]
[tex]\displaystyle f).4+8+12+...+2000=4(1+2+3+...+500)= \\ \\ =4 \times \frac{500(500+1)}{2} =4 \times \frac{500 \times 501}{2} =4\times \frac{250500}{2} = \\ \\ =4 \times 125250=501000[/tex]
[tex]\displaystyle g).3+7+11+15+...+43 \\ 3=1 \times 4-1 \\ 7=2 \times 4-1 \\ 11=3 \times 4-1 \\ 15=4 \times 4-1 \\ . \\ . \\ . \\ 43=4 \times 11-1 \\ 3+7+11+15+...+43=4(1+2+3+4+...+11)-1 \times 11 \\ \\ 3+7+11+15+...+43= 4 \times \frac{11 \times 12}{2} -11 \\ \\ 3+7+11+15+...+43=4 \times \frac{132}{2} -11 \\ \\ 3+7+11+15+...+43=4 \times 66-11 \\ \\ 3+7+11+15+...+43=264-11 \\ \\ 3+7+11+15+...+43=253[/tex]
