1) (14＋2010)＋(13＋2010×2）+（12+2010×3）+……+（2+2010×13）+（1+2010×14）
=(14+13+...+2+1)+2010(1+2+3+...+14)
=2011*(1+2+3+4+...+14）
=2011*[(1+14）*14/2]
=211155
2) 1/(3+6)+1/(3+6+9)+.+1/(3+6+9+...+150)
=1/(3+3*2)+1/(3+3*2+3*3)+.+1/(3+3*2+3*3+...+3*50)
=(1/3)*[1/(1+2)+1/(1+2+3)+.+1/(1+2+3+...+50)]
=(1/3)*{1/3+2/[3*(1+3)]+2/[4*(1+4)]+2/[5*(1+5)]+2/[6*(1+6)]+.+2/[50*(1+50)]}
=(1/3)*(2/2*3+2/3*4+2/4*5+2/5*6+2/6*7+.+2/50*51)
=(2\3)*(1/2*3+1/3*4+1/4*5+1/5*6+1/6*7+.+1/50*51)
=(2\3)*[(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+(1/6-1/7)+.+(1/50-1/51)]
=(2/3)*(1/2-1/51)
=49\153
3)我認(rèn)為第五個(gè)數(shù)為11/30比較合理.
4/3-15/56+9/20-7/12-11/30+13/42
=4/3-7/12+9/20-11/30+13/42-15/56
=4/3-7/3*4+9/4*5-11/5*6+13/6*7-15/7*8
=4/3-(7/3-7/4)+(9/4-9/5)-(11/5-11/6)+(13/6-13/7)-(15/7-15/8)
=4/3-7/3+7/4+9/4-9/5-11/5+11/6+13/6-13/7-15/7+15/8
=-1+4-4+4-4+15/8
=7/8
如果第五個(gè)數(shù)為11/33 則有
4/3-15/56+9/20-7/12-11/33+13/42
=4/3-7/12+9/20-11/33+13/42-15/56
=4/3-7/12+9/20-10/30+13/42-15/56
=4/3-7/3*4+9/4*5-10/5*6+13/6*7-15/7*8
=4/3-(7/3-7/4)+(9/4-9/5)-(10/5-10/6)+(13/6-13/7)-(15/7-15/8)
=4/3-7/3+7/4+9/4-9/5-10/5+10/6+13/6-13/7-15/7+15/8
=-1+4-19/5+23/6-4+15/8
=-1+4-4+4-4+15/8+1/5-1/6
=109/120