作业帮观察下列式子1-1/4^2=15/16,(1-1/4^2)(1-1/5^2)=9/10;(1-1/4^2)(1-1/5^2)(1-1/6^2)=7/8…;;则式子(1-1/4^2)(1-1/5^2)(1-1/6^2)…[1-1/(n-1)^2](1-1/n^2)=?
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/14 10:19:53
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观察下列式子1-1/4^2=15/16,(1-1/4^2)(1-1/5^2)=9/10;(1-1/4^2)(1-1/5^2)(1-1/6^2)=7/8…;;则式子(1-1/4^2)(1-1/5^2)(1-1/6^2)…[1-1/(n-1)^2](1-1/n^2)=?
观察下列式子1-1/4^2=15/16,(1-1/4^2)(1-1/5^2)=9/10;(1-1/4^2)(1-1/5^2)(1-1/6^2)=7/8…;;则式子(1-1/4^2)(1-1/5^2)(1-1/6^2)…[1-1/(n-1)^2](1-1/n^2)=?
观察下列式子1-1/4^2=15/16,(1-1/4^2)(1-1/5^2)=9/10;(1-1/4^2)(1-1/5^2)(1-1/6^2)=7/8…;;则式子(1-1/4^2)(1-1/5^2)(1-1/6^2)…[1-1/(n-1)^2](1-1/n^2)=?
多次运用平方差公式
(1-1/4^2)(1-1/5^2)(1-1/6^2)…[1-1/(n-1)^2](1-1/n^2)
=(1-1/4)(1+1/4)(1-1/5)(1+1/5)...(1-1/(n-1))(1+1/(n-1))(1-1/n)(1+1/n)
=3/4*5/4*4/5*6/5...(n-2)/(n-1)*n/(n-1)*(n-1)/n*(n+1)/n
=3/4*(n+1)/n
=3(n+1)/4n
1-1/4^2=15/16,
(1-1/4^2)(1-1/5^2)=9/10;
(1-1/4^2)(1-1/5^2)(1-1/6^2)=7/8
…;;
则式子(1-1/4^2)(1-1/5^2)(1-1/6^2)…[1-1/(n-1)^2](1-1/n^2)=3(n+1)/4n
(1-1/4^2)(1-1/5^2)(1-1/6^2)…[1-1/(n-1)^2...
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1-1/4^2=15/16,
(1-1/4^2)(1-1/5^2)=9/10;
(1-1/4^2)(1-1/5^2)(1-1/6^2)=7/8
…;;
则式子(1-1/4^2)(1-1/5^2)(1-1/6^2)…[1-1/(n-1)^2](1-1/n^2)=3(n+1)/4n
(1-1/4^2)(1-1/5^2)(1-1/6^2)…[1-1/(n-1)^2](1-1/n^2)
=(1-1/4)(1+1/4)(1-1/5)(1-1/5)... [ 1-1/(n-1) ][1+1/(n-1) ] [ 1-1/n] [ 1+1/n]
=3/4×5/4×4/5×6/5×...×(n-2)/(n-1)×n/(n-1)×(n-1)/n×(n+1)/n
=3/4 × (n+1)/n
=3(n+1)/4n
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