5/(2×3×4)+7/(3×4×5)+……+17/(8×9×10)+19/(9×10×11)
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5/(2×3×4)+7/(3×4×5)+……+17/(8×9×10)+19/(9×10×11)
∵5/(2×3×4)=(1/4)(1/2+1/3)=(1/2)(1/2-1/4)+1/3-1/4,
7/(3×4×5)=(1/5)(1/3+1/4)=(1/2)(1/3-1/5)+1/4-1/5,
9/(4×5×6)=(1/6)(1/4+1/5)=(1/2)(1/4-1/6)+1/5-1/6,
11/(5×6×7)=(1/7)(1/5+1/6)=(1/2)(1/5-1/7)+1/6-1/7,
13/(6×7×8)=(1/8)(1/6+1/7)=(1/2)(1/6-1/7)+1/7-1/8,
15/(7×8×9)=(1/9)(1/7+1/8)=(1/2)(1/7-1/9)+1/8-1/9,
17/(8×9×10)=(1/10)(1/8+1/9)=(1/2)(1/8-1/10)+1/9-1/10,
19/(9×10×11)=(1/11)(1/9+1/10)=(1/2)(1/9-1/11)+1/10-1/11.
∴将上述8个等式左右分别相加,得:
5/(2×3×4)+7/(3×4×5)+9/(4×5×6)+······+19/(9×10×11)
=(1/2)(1/2+1/3-1/10-1/11)+1/3-1/11
=1/4+(1/3)(1/2+1)-1/10-(1/11)(1/2+1)
=1/4+1/2-1/10-3/22
=3/4-1/10-3/22
=(1/2)(3/2-1/5-3/11)
=(1/2)[(15-2)/(2×5)-3/11]
=(1/2)(13/10-3/11)
=(1/2)(143-30)/110
=113/220.
7/(3×4×5)=(1/5)(1/3+1/4)=(1/2)(1/3-1/5)+1/4-1/5,
9/(4×5×6)=(1/6)(1/4+1/5)=(1/2)(1/4-1/6)+1/5-1/6,
11/(5×6×7)=(1/7)(1/5+1/6)=(1/2)(1/5-1/7)+1/6-1/7,
13/(6×7×8)=(1/8)(1/6+1/7)=(1/2)(1/6-1/7)+1/7-1/8,
15/(7×8×9)=(1/9)(1/7+1/8)=(1/2)(1/7-1/9)+1/8-1/9,
17/(8×9×10)=(1/10)(1/8+1/9)=(1/2)(1/8-1/10)+1/9-1/10,
19/(9×10×11)=(1/11)(1/9+1/10)=(1/2)(1/9-1/11)+1/10-1/11.
∴将上述8个等式左右分别相加,得:
5/(2×3×4)+7/(3×4×5)+9/(4×5×6)+······+19/(9×10×11)
=(1/2)(1/2+1/3-1/10-1/11)+1/3-1/11
=1/4+(1/3)(1/2+1)-1/10-(1/11)(1/2+1)
=1/4+1/2-1/10-3/22
=3/4-1/10-3/22
=(1/2)(3/2-1/5-3/11)
=(1/2)[(15-2)/(2×5)-3/11]
=(1/2)(13/10-3/11)
=(1/2)(143-30)/110
=113/220.
简便计算:1155*(5/2*3*4+7/3*4*5+…+17/8*9*10+19/9*10*11)
简便计算:5/2*3*4+7/3*4*5+…+17/8*9*10+19/9*10*11
1 1 2 3 3 5 4 7 5 9 6 11 7 13 8 15 9 17 10 19 ….….N 用第N个数表示“
1 2 6 7 15 16…… 3 5 8 14 17…… 4 9 13 18…… 10 12 19…… 11 20……
5/2*3*4+7/3*4*5+...+8*9*10/17+19/9*10*11求简便运算
5/2×3×4+7/3×4×5+.+17/8×9×10+19/9×10×11怎么求
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+……1
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21……+99+1
1+2-3+4-5+6-7+8-9+10-11+12-13+14-15+16-17+18-19+20……-99+100
(1,2);(3,4,5,6);(7,8,9,10,11,12);(13,14,15,16,17,18,19,20)……
1-2-3+4+5-6-7+8+9-10-11+12+……+2009-2010
1+2-3-4+5+6-7-8+9+10-11-12+……+2001=?