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Question
Show that `(a-b)^2 , (a^2 + b^2 ) and ( a^2+ b^2) ` are in AP.
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Solution
The given numbers are `(a-b)^2 , (a^2 + b^2 ) and ( a+ b^2) `
Now,
`(a^2 + b^2 ) - (a-b)^2 = a^2 + b^2 - (a^2 -2ab + b^2 ) = a^2 + b^2 - a^2 + 2ab - b^2 = 2ab`
`(a+b)^2 - (a^2 + b^2) = a^2 + 2ab + b^2 - a^2 - b^2 = 2ab`
So, `(a^2 + b^2 ) - (a -b)^2 = (a + b)^2 - (a^2 + b^2 ) =2ab ` (Constant)
Since each term differs from its preceding term by a constant, therefore, the given numbers are in AP.
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