Question

1/1+2 +1/1+2+3 +1/1+2+3+4……+1/1+2+3+……+99=

1/1+2 +1/1+2+3 +1/1+2+3+4……+1/1+2+3+……+99=

(1+1/2)×（1+1/4）×(1+1/6)×……×(1+1/10)×(1-1/3)×(1-1/5)×……×(1-1/9)

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Answer 1

1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+…+1/(1+2+3+…+98)+1/(1+2+3+…+99)
=1/[（1+2）*2/2]+1/[（1+3）*3/2]+1/[(1+4)*4/2]+……+1/[（1+99）*99/2]
=2/(2*3)+2/(3*4)+2/4*5+……+2/（100*99）
=[1/2*3+1/3*4+1/4*5+......+1/99*100]*2
=（1/2-1/3+1/3-1/4+1/4-1/5+……+1/99-1/100）*2
=(1/2-1/100)*2
=49/100*2
=49/50
原因：1/(1+2)=2*(1/2-1/3)
1/(1+2+3)=2*(1/3-1/4)
1/(1+2+3+4)=2*(1/4-1/5)
………………………………
1/(1+2+……+k)=2*【1/k-1/(1+k)】
…………………
1/（1+2+3+...+99）=2*（1/99-1/100)
1/（1+2）+1/（1+2+3）+1/（1+2+3+4）+......+1/（1+2+3+...+99)=2*（1/2-1/3+1/3-1/4+1/4-1/5+……+1/k-1/(1+k）+……+1/99-1/100）=2*（1/2-1/100)=49/50=0.98

(1+1\2)*(1+1\4）*（1+1\6)*...*(1+1\10)*(1-1\3)*(1-1\5)*...*(1-1\9)
=(1+1\2)*(1-1\3)*(1+1\4）*（1-1\5)*(1+1\6)*(1-1/7)*(1+1/8)*(1-1\9)
*(1+1\10)
=1*1*1*1*(1+1\10)
=1+1/10
=11/10
原因：(1+1/2)*(1+1/4)*(1+1/6)*(1+1/8)*(1+10/1)*(1-1/3)*(1-1/5)*(1-1/7)*(1-9/1)
=3/2*5/4*....*11/10*2/3*4/5.....*8/9
=3/2*2/3*5/4*4/5*.....*9/8*8/9*11/10
=1*1*...*1*11/10
=11/10

Answer 2

(1)∵1+2=3*2/2,1+2+3=3*4/2,1+2+3+4=4*5/2~
∴原式=2/2*3+2/3*4+2/4*5+~+2/99*100
=2(1/2*3+1/3*4+~+1/99*100)
=2(1/2-1/3+1/3-1/4+~+1/99-1/100) [1/n*(n+1)=1/n-1/n-1]
=2(1/2-1/100)
=49/50
(2)原式=(3/2)×（5/4）×(7/6)×……×(11/10)×(2/3)×(4/5)×……×(8/9)
=11/10

Answer 3

解：1、1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+…+1/(1+2+3+…+98)+1/(1+2+3+…+99)
=1/[（1+2）*2/2]+1/[（1+3）*3/2]+1/[(1+4)*4/2]+……+1/[（1+99）*99/2]
=2/(2*3)+2/(3*4)+2/4*5+……+2/（100*99）
=[1/2*3+1/3*4+1/4*5+......+1/99*100]*2
=（1/2-1/3+1/3-1/4+1/4-1/5+……+1/99-1/100）*2
=(1/2-1/100)*2
=49/100*2
=49/50
2、1+1\2)*(1+1\4）*（1+1\6)*...*(1+1\10)*(1-1\3)*(1-1\5)*...*(1-1\9)
=(1+1\2)*(1-1\3)*(1+1\4）*（1-1\5)*(1+1\6)*(1-1/7)*(1+1/8)*(1-1\9)
*(1+1\10)
=1*1*1*1*(1+1\10)
=1+1/10
=11/10

Answer 4

(1)
注意到2/(n(n+1))=2/n-2/(n+1)
所以1/1+2 +1/1+2+3 +1/1+2+3+4……+1/1+2+3+……+99
=(2/1-2/2)+(2/2-2/3)+...+(2/99-2/100)
=2/1-2/100
=2-1/50
=99/50

(2)
1+1/2n=(2n+1)/2n
1-1/(2n+1)=2n/(2n+1)
所以(1+1/2)×（1+1/4）×(1+1/6)×……×(1+1/10)×(1-1/3)×(1-1/5)×……×(1-1/9)
=3/2*5/4*...*11/10*2/3*4/5*...*8/9
=11/10