Question

简算：【提示：平方差公式：a*a-b*b=(a+b)*(a-b)】谢谢！！！

一、1/(2*2-1)+1/(4*4-1)+1/(6*6-1)+.........+1/(100*100-1)

二、【1-1/（2*2）】*【1-1/(3*3）】*【1-1/(4*4）】*.........*【1-1/（2004*2004）】

Answer 1

一、1/(2*2-1)+1/(4*4-1)+1/(6*6-1)+.........+1/(100*100-1)
=1/(2+1)(2-1)+1/(4+1)(4-1)+...+1/(100+1)(100-1)
=(1/2)(1/(2-1)-1/(2+1)+1/(4-1)-1/(4+1)+...+1/(100-1)-1/(100+1))
=(1/2)(1-1/3+1/3-1/5+1/5-1/7+...+1/99-1/101)
=(1/2)(1-1/101)
=50/101;
二、【1-1/（2*2）】*【1-1/(3*3）】*【1-1/(4*4）】*.........*【1-1/（2004*2004）】
=(1+1/2)(1-1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)*....*(1+1/2004)(1-1/2004)
=(3/2)(1/2)(4/3)(2/3)(5/4)(3/4)*..*(2005/2004)(2003/2004)
=(2005/2)(1/2004)
=2005/4008

Answer 2

1/(2*2-1)+1/(4*4-1)+1/(6*6-1)+.........+1/(100*100-1)
=1/((2+1)(2-1))+1/((4+1)(4-1))+1/((6+1)(6-1)........+1/((100+1)(100-1))
=1/(3*1)+1/(5*3)+1/(7*5)+......+1/(101*99)
=1/2(1/1-1/3)+1/2(1/3-1/5)+1/2(1/5-1/7)+........+1/2(1/99-1/101)
=1/2(1-1/99)
=49/99

【1-1/（2*2）】*【1-1/(3*3）】*【1-1/(4*4）】*.........*【1-1/（2004*2004）】
=(1+1/2)(1-1/2)*(1+1/3)(1-1/3)*(1+1/4)(1-1/4)...........*(1+1/2004)(1-1/2004)
=3/2*1/2*4/3*2/3*5/4*3/4*........*2005/2004*2003/2004
=1/2*2005/2004=2005/4008

Answer 3

解：一、1/(2*2-1)+1/(4*4-1)+1/(6*6-1)+.........+1/(100*100-1)
=1/(1×3)+1/(3×5)+1/(5×7)+1/(7×9)+.....+1/(99×101)
=1/2×(1-1/3+1/3-1/5+1/5-1/7+1/7-1/9+.....+1/99-1/101)
=1/2×(1-1/101)
=1/2×100/101
=50/101
二 、【1-1/（2*2）】*【1-1/(3*3）】*【1-1/(4*4）】*.........*【1-1/（2004*2004）】
=(1+1/2)(1-1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)*.......*(1+1/2004)(1-2004)
=3/2×1/2×4/3×2/3×5/4×3/4×........×2005/2004×2003/2004
=(3/2×4/3×5/4×6/5×7/6×2005/2004)×(1/2×2/3×3/4×4/5×5/6×......×2003/2004)
=2005/2×1/2004
=2005/2008
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Answer 4

平方差公式是
(a+b)(a-b)=a^2-b^2
（a-b)(-a-b)
(把后面一个式子的负号提出来就可以了）
=
-（a-b)(a+b)
=
-
(a^2-b^2)
=b^2-a^2
答案一定是对的
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