Question

Calculati
a) 9*{8+7*[6+5*(4+3*2*1)]}:(9+8+7+...2+1)
b) 150+151+...+2013
c) 37+38+39+...+256
d) 2+4+6+...+2012
e) 1+3+5+7+...+2013

Answer 1

a)9*{8+7*[6+5*(4+3*2*1)]}:(9+8+7+...2+1)=
=9*{8+7*[6+5*(4+6)]}:(9*10)/2=
=9*[8+7*(6+5*10)]:(9*10)/2=
=9*(8+7*56):(9*10)/2=
=9*400:(9*10)/2=
=9*400*2/(9*10)=(9 cu 9 se reduc, 400 se simplifica cu 10)
=40*2=80

b) 150+151+...+2013=
=(1+2+3+...+2013)-(1+2+3+...+149)=
=(2013*2014)/2-(149*150)/2=
=2013*1007-149*75=2015916

c)37+38+39+...+256=
=(1+2+3+...+256)-(1+2+3+...+36)=
=(256*257)/2-(36*37)/2=
=125*257-18*37=
=31459

d)2+4+6+...+2012=
=2(1+2+3+...+1006)=
=2*1006*1007/2=
=1006*1007=1 013 042

e)1+3+5+7+...+2013=
=(2*0+1)+(2*1+1)+(2*2+1)+...+(2*1006+1)=
=2*0+2*1+2*2+...+2*1006+1+1+1+..+1+1(de 1007 ori)=
=2(0+1+2+3+...+1006)+1007=
=2*1006*1007/2+1007=
=1006*1007+1007=
=1007(1006+1)=
=1007*1007=1007²=1014049