Question

Să se calculeze:
1+2+3+....+2005=
2+4+6+8+....2010=
1+3+5+7+....1009=
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Answer 1

1+2+3+....+2005=suma Gauss=(2005×2005+1)]:2=(2005×2006):2=2011015

1+2+3+....+2005=2011015

2+4+6+8+....2010=2×(1+2+3+4+.......1005)=2 ×suma gauss
=2×(1005×1006):2
2+4+6+8+....2010=1005×1006=1011030

2+4+6+8+....2010=1011030


1+3+5+7+....1009=
progresie din 2 in 2.....unde a1=1, an=1009 si ratia este 2( cresc din 2 in 2)
S=[(a1+an)×n]:2
n-nr de termeni
an=a1+(n-1)×r
1009=1+(n-1)×2
1009=1+2n-2
1009+1=2n
1010=2n
n=505

S=[(a1+an)×n]:2=[(1+1005)×505]:2=254015

1+3+5+7+....1009 = 254015

Answer 2

1+2+3++...2005=
2005x2006:2=2011015

2+4+....2010=
2x(1+2+3+++1005)x1006:2=
2x1005x1006:2=1011030

1+3+5++1009=
2x(1+2+++505)x506:2=
2x505x506:2=255530