观察下列各式 1/6=1/2×3=1/2-1/3 1/12=1/3×4=1/3-1/4 1/20=1/4×5=1/4-1
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观察下列各式 1/6=1/2×3=1/2-1/3 1/12=1/3×4=1/3-1/4 1/20=1/4×5=1/4-1/5
(1)猜想:1/n(n+1)=___________
(2)化简:1/(n+1)(n+2)+1/(n+2)(n+3)+1/(n+3)(n+4)
(1)猜想:1/n(n+1)=___________
(2)化简:1/(n+1)(n+2)+1/(n+2)(n+3)+1/(n+3)(n+4)
(1) 规律:1/m*(m+1) = 1/m - 1/(m+1) ; m = 1,2,3,...证明:右式 = 1/m - 1/(m+1) = (m+1)/m*(m=1) - m/m*(m+1) = 1/m*(m+1) = 左式;此规律实际上说的是:如果一个分数分子是1,分母可以写成两个连续整数的乘积m与m+1,那么这个分数就可以写成1/m - 1/(m+1); (2) 由上述规律有:根据上述规律再结合该题;分母为相差1的两个数相乘的式子容易展开,即 1/(x-2)(x-3) = 1/(x-3) - 1/(x-2); 1/(x-1)(x-2) = 1/(x-2) - 1/(x-1); 而分母相差不是1的就需要变下型,考虑到这里分母相差2,那么我们只要在每个因式上提出一个2,那么就符合条件了,如下:1/(x-1)(x-3) = 1/4 * 1/ (x/2-1/2)(x/2-3/2) = 1/4 * ( 1/(x/2 - 3/2) - 1/(x/2-1/2) ) = 1/2 * ( 1/(x-3) - 1/(x-1) ); 1/n(n+1)=1/n-1/(n+1)
化简=(n+3)(n+4)+(n+1)(n+4)+(n+1)(n+2)/(n+1)(n+2)(n+3)(n+4)
=(n+4)(n+3+n+1)+(n+1)(n+2)/(n+1)(n+2)(n+3)(n+4)
=(n+4)2(n+2)+(n+1)(n+2)/(n+1)(n+2)(n+3)(n+4)
=3(n+3)(n+2)+(n+1)(n+2)/(n+1)(n+2)(n+3)(n+4)
(1)猜想:1/n(n+1) = 1/n-1/(n+1)
(2)化简:1/(n+1)(n+2)+1/(n+2)(n+3)+1/(n+3)(n+4)
=1/(n+1)-1/(n+4)
=3/(n+1)(n+4)
化简=(n+3)(n+4)+(n+1)(n+4)+(n+1)(n+2)/(n+1)(n+2)(n+3)(n+4)
=(n+4)(n+3+n+1)+(n+1)(n+2)/(n+1)(n+2)(n+3)(n+4)
=(n+4)2(n+2)+(n+1)(n+2)/(n+1)(n+2)(n+3)(n+4)
=3(n+3)(n+2)+(n+1)(n+2)/(n+1)(n+2)(n+3)(n+4)
(1)猜想:1/n(n+1) = 1/n-1/(n+1)
(2)化简:1/(n+1)(n+2)+1/(n+2)(n+3)+1/(n+3)(n+4)
=1/(n+1)-1/(n+4)
=3/(n+1)(n+4)
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