Question

Calculati:
a)8+16+24+32+....................+4000=
b)11+22+33+44+....................+7700=
c)21+42+63+84+...................+2100=
d)4+8+12+16+..............+400=
e)25+50+75+100+........400=
f)10+20+30+40+............+480=
g)6+12+18+24+.............1200=
h)30+60+90+120+.............900=
i)15+30+45+60+.........+600=
VA ROG EXPLICATI-MI CUM A-TI CALCULAT

Answer 1

$\displaystyle a).8+16+24+32+...+4000=8(1+2+3+...+500)= \\ \\ =8 \cdot \frac{500(500+1)}{2} =8 \cdot \frac{500 \cdot 501}{2} =8 \cdot \frac{250500}{2} =8 \cdot 125250=1002000$

$\displaystyle b).11+22+33+44+...+7700=11(1+2+3+...+700)= \\ \\ =11 \cdot \frac{700(700+1)}{2} =11 \cdot \frac{700 \cdot 701}{2} =11 \cdot \frac{490700}{2} = \\ \\ =11 \cdot 245350=2698850$

$\displaystyle c).21+42+63+84+...+2100=21(1+2+3+...+100)= \\ \\ =21 \cdot \frac{100(100+1)}{2} =21 \cdot \frac{100 \cdot 101}{2} =21 \cdot \frac{10100}{2} =21 \cdot 5050=106050$

$\displaystyle d).4+8+12+16+...+400=4(1+2+3+...+100)= \\ \\ =4 \cdot \frac{100(100+1)}{2} =4 \cdot \frac{100 \cdot 101}{2} =4 \cdot \frac{10100}{2} =4 \cdot 5050=20200$

$\displaystyle e).25+50+75+100+...+400=25(1+2+3+...+16)= \\ \\ =25 \cdot \frac{16(16+1)}{2} =25 \cdot \frac{16 \cdot 17}{2} =25 \cdot \frac{272}{2} =25 \cdot 136=3400$

$\displaystyle f).10+20+30+40+...+480=10(1+2+3+...+48)= \\ \\ =10 \cdot \frac{48(48+1)}{2} =10 \cdot \frac{48 \cdot 49}{2} =10 \cdot \frac{2352}{2} =10 \cdot 1176=11760$

$\displaystyle g).6+12+18+24+...+1200=6(1+2+3+...+200)= \\ \\ =6 \cdot \frac{200(200+1)}{2} =6 \cdot \frac{200 \cdot 201}{2} =6 \cdot \frac{40200}{2} =6 \cdot 20100=120600$

$\displaystyle h).30+60+90+120+...+900=30(1+2+3+...+30)= \\ \\ =30 \cdot \frac{30(30+1)}{2} =30 \cdot \frac{30 \cdot 31}{2} =30 \cdot \frac{930}{2} =30 \cdot 465=13950$

$\displaystyle i).15+30+45+60+...+600=15(1+2+3+...+40)= \\ \\ =15 \cdot \frac{40(40+1)}{2} =15 \cdot \frac{40 \cdot 41}{2} =15 \cdot \frac{1640}{2} =15 \cdot 820=12300$