Toán Lớp 7: 1/2(1+1/1.3)(1+1/2.4)(1+1/3.5)…..(1+1/2019.2021)
nhanh ạ
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Toán Lớp 7: 1/2(1+1/1.3)(1+1/2.4)(1+1/3.5)…..(1+1/2019.2021)
nhanh ạ
Comments ( 1 )
1/2 . ( 1 + 1/( 1 . 3 ) ) . ( 1 + 1/( 2 . 4 ) ) . ( 1 + 1/( 3 . 5 ) ) ….. ( 1 + 1/( 2019 . 2021 ) )
= 1/2 . ( ( 1 . 3 ) + 1 )/( 1 . 3 ) . ( ( 2 . 4 ) + 1 )/( 2 . 4 ) . ( 3 . 5 + 1 )/( 3 . 5 ) …. ( 2019 . 2021 + 1 )/( 2019 . 2021 )
Ta có : n . ( n + 2 ) + 1 = n^2 + 2n + 1 = n^2 + 2 . n . 1 + 1^2 = ( n + 1 )^2
Do đó : 1/2 . ( ( 1 . 3 ) + 1 )/( 1 . 3 ) . ( ( 2 . 4 ) + 1 )/( 2 . 4 ) . ( 3 . 5 + 1 )/( 3 . 5 ) …. ( 2019 . 2021 + 1 )/( 2019 . 2021 )
= 1/2 . ( 1 + 1 )^2/( 1 . 3 ) . ( 2 + 1 )^2/( 2 . 4 ) . ( 3 + 1 )^2/( 3 . 5 ) …. ( 2019 + 1 )^2/( 2019 . 2021 )
= 1/2 . 2^2/( 1. 3 ) . 3^2/( 2 . 4 ) . 4^2/( 3 . 5 ) …. ( 2020 )^2/( 2019 . 2021 )
= 1/2 . ( 2^2 . 3^2 . 4^2 …. 2020^2 )/( 1 . 3 . 2 . 4 . 3 . 5 …. 2019 . 2021 )
= 1/2 . ( ( 2 . 3 . 4 …. 2020 ) . ( 2 . 3 . 4 … 2020 ) )/( ( 1 . 2 . 3 …. 2019 ) . ( 3 . 4 . 5 … 2021 ) )
= 1/2 . ( 2020 . 2 )/( 1 . 2021 )
= ( 2020 . 2 )/( 2021 . 2 )
= 2020/2021