Question

CALCULATI URMATOARELE SUME
A) S=1+2+3+.....+1000+1001
B)S= 1+3+5+7+....+97+99
C)S= 2+4+6+......+100+102
D)S=1+5+9+13+.......+141+145
E)S= 11+22+33+44+...+242+253
F)S= 9+46+83+120+....+1119+1156
G)S=203+303+403+....+10003+10103
H)S=1007+1005+1003+....+117+115
I)S=9+19+29+....+1999
J)S=3+5+7+....+20001-2-4-6-....-2000
K)S=100.99-99.98+98.97-97.96+....+4.3-3.2+2.1

Answer 1

A)  S=1+2+3+.....+1000+1001=1001*1002/2=501501

B)  S= 1+3+5+7+....+97+99= (99+1)*50/2=2500
2n-1=99
2n=100
n=50

C) S= 2+4+6+......+100+102= 2*(1+2+3+....+51)=2*51*52/2=2652

D) S=1+5+9+13+.......+141+145= (145+1)*37/2=2701
a=4*n-3
4*n-3=145
4*n=148  :4
n=37

E)S= 11+22+33+44+...+242+253=11*(1+2+3+...23)=11*23*24/2=3036

F)S= 9+(46+83+120+....+1119+1156)=
=9+(1156+46)*31/2=9+18631=18640
a₁=9
a₂=9+37*1
a₃=9+37*2
9+37*n=1156
37n=1156-9
37n=1147  :37
n=31

G)S=203+303+403+....+10003+10103=
= 2*100+3+3*100+3+4*100+3.....+11*100+3=
=100*(2+3+...+11)+11*3=100*(13*10/2)+33=6500+33=6533

H)S=1007+1005+1003+....+117+115=
=115+117+.....1007=(1007+115)*447/2=1122*447/2=250767

an=2n-1
2n-1=115
2n=116  :2
n=58
2n-1=1007
2n=1008
n=504
504-57=447 termeni

I)  S=9+19+29+....+1999=9+(9+10+9+20+9+30...+9+1190)=
=10*(1+2+3+...119)+ 119*9=10*119*120/2+119*9=71400+1071=7247

J)S=3+5+7+....+2001                        2n-1=2001; n=1001
       -2-4-6-......-2000
___________________ =1*1001 -1 =1000
       1+1+1+...........+1


K)S=100.99-99.98+98.97-97.96+....+4.3-3.2+2.1 =
=1.01+1.01+1.01+....1.01= 1.01*100= 101