裂项的练习题

问题描述:

裂项的练习题
有什么练习册里面有这类题。

(1)1x2+2x3+3x4+4x5+...+2002x2003
=1/3*1*2*3+1/3[2*3*4-1*2*3]+1/3[3*4*5-2*3*4]+.+1/3[2002*2003*2004-2001*2002*2003]
=1/3*2002*2003*2004
=2678684008
(2)(1-1/2^2)*(1-1/3^2)*(1-1/4^2)……(1-1/2010^2)
=(1+1/2)(1-1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)×……×(1+1/2010)(1-1/2010)
=3/2×4/3×5/4×……×2011/2010×1/2×2/3×3/4×……×2009/2010
=2011/2×1/2010
=2011/4020
(3)1/1*3+1/2*4+1/3*5+1/4*6+1/5*7.1/98*100+1/99*101
=(1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+1/5-1/7+……+1/98-1/100+1/99-1/101)÷2
=(1+1/2-1/100-1/101)÷2
=15049/10100÷2
=15049/20200
(4)1/2+1/(2+4)+1/(2+4+6)+…1/(2+4+6+…+200)
=1/(1×2)+1/(2×3)+1/(3×4)+1/(100×101)
=1-1/2+1/2-1/3+1/3-1/4+……+1/100-1/101)
=1-1/101
=100/101
(5)6分之1+12分之1+24分之1+48分之1+96分之1+192分之1
=1/6×(1+1/2+1/4+1/8+1/16+1/32)
=1/6×(1-1/32)
=1/6-1/192
=31/192
(6)(2²+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)
=(2^2-1)(2²+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)/(2^2-1)
=(2^4-1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)/3
= (2^8-1)(2^8+1)(2^16+1)(2^32+1)/3
=(2^16-1)(2^16+1)(2^32+1)/3
=(2^32-11)(2^32+1)/3
=(2^64-1)/3
(7)1+1/(1+2)+1/(1+2++3)+……+1/(1+2+3+……+n)
=2*【1/2+1/2*(1+2)+1/2*(1+2++3)+……+1/2*(1+2+3+……+n)】
=2*【1-1/2+1/2-1/3+1/3-1/4+……+1/n-1/(n+1)】
=2*【1-1/(n+1)】
=2n/(n+1)