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If a ,\ b ,\ c ,\ d are in G.P., show th...

If `a ,\ b ,\ c ,\ d` are in G.P., show that: `(a b+b c+c d)^2=(a^2+b^2+c^2)(b^2+c^2+d^2)`

Text Solution

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Let r be the common ratio of the G.P., a,b,c,d. Then
b=ar,c`=ar^(2)andd=ar^(3)`
`L.H.S=(ab+bc+cd)^(2)`
`(aar+arar^(2))+ar^(2)ar^(3))^(2)`
`=a^(4)r^(2)(1+r^(2)+r^(4))^(2)`
`R.H.S=(a^(2)+b^(2)+c^(2))(b^(2)+c^(2)+d^(2))`
`=(a^(2)+a^(2)r^(2)+a^(2)r^(4))(a^(2)r^(2)+a^(2)r^(4)+a^(2)r^(6))`
`=a^(2)(1+r^(2)+r^(4))a^(2)r^(2)(1+r^(2)+r^(4))`
`=a^(4)r^(2)(1-r^(2)+r^(4))^(2)`
`therefore` L.H.S=R.H.S
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