5/2×3×4+7/3×4×5+.+17/8×9×10+19/9×10×11怎么求2013某工大附中入学题

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5/2×3×4+7/3×4×5+.+17/8×9×10+19/9×10×11怎么求2013某工大附中入学题
5/2×3×4+7/3×4×5+.+17/8×9×10+19/9×10×11怎么求
2013某工大附中入学题

5/2×3×4+7/3×4×5+.+17/8×9×10+19/9×10×11怎么求2013某工大附中入学题
∵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.