观察下列等式:1x1/2=1-1/2,2x2/3=2-2/3,3x3/4=3-3/4,……
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观察下列等式:1x1/2=1-1/2,2x2/3=2-2/3,3x3/4=3-3/4,……
1.猜想并写出第n个等式.
2.请你计算:(1-1/2)/10x(2-2/3)/9x(3-3/4)/8x.x(10-10/11)/1
1.猜想并写出第n个等式.
2.请你计算:(1-1/2)/10x(2-2/3)/9x(3-3/4)/8x.x(10-10/11)/1
解1:
第n个等式是:n×(n/(n+1)=n-n/(n+1)
解2:
[(1-1/2)/10]×[(2-2/3)/9]×[(3-3/4)/8]×……×[(10-10/11)/1]
=[(1×1/2)/10]×[(2×2/3)/9]×[(3×3/4)/8]×……×[(10×10/11)/1]
=[(1×1/2)/1][(2×2/3)/2][(3×3/4)/3]×……×[(10×10/11)/10]
=(1/2)[(2×2/3)/2][(3×3/4)/3]×……×[(10×10/11)/10]
=(1/3)[(3×3/4)/3]×……×[(10×10/11)/10]
=(1/4)[(4×4/5)/4]×……×[(10×10/11)/10]
=(1/5)[(5×5/6)/5]×……×[(10×10/11)/10]
=(1/6)[(6×6/7)/6]×……×[(10×10/11)/10]
=(1/7)[(7×7/8)/7]×……×[(10×10/11)/10]
=(1/8)[(8×8/9)/8]×……×[(10×10/11)/10]
=(1/9)[(9×9/10)/9]×[(10×10/11)/10]
=(1/10)[(10×10/11)/10]
=1/11
第n个等式是:n×(n/(n+1)=n-n/(n+1)
解2:
[(1-1/2)/10]×[(2-2/3)/9]×[(3-3/4)/8]×……×[(10-10/11)/1]
=[(1×1/2)/10]×[(2×2/3)/9]×[(3×3/4)/8]×……×[(10×10/11)/1]
=[(1×1/2)/1][(2×2/3)/2][(3×3/4)/3]×……×[(10×10/11)/10]
=(1/2)[(2×2/3)/2][(3×3/4)/3]×……×[(10×10/11)/10]
=(1/3)[(3×3/4)/3]×……×[(10×10/11)/10]
=(1/4)[(4×4/5)/4]×……×[(10×10/11)/10]
=(1/5)[(5×5/6)/5]×……×[(10×10/11)/10]
=(1/6)[(6×6/7)/6]×……×[(10×10/11)/10]
=(1/7)[(7×7/8)/7]×……×[(10×10/11)/10]
=(1/8)[(8×8/9)/8]×……×[(10×10/11)/10]
=(1/9)[(9×9/10)/9]×[(10×10/11)/10]
=(1/10)[(10×10/11)/10]
=1/11
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