1.If a,b and c are in ap ,show that bc-a2 ,ca-b2 and ab-c2 are in A.P.
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Given that:-
a,b,c are in A.P
D = b-a = c-b
=> 2b = a+c-----------(1)
In A.P
D = (ca - b²) - (bc - a²) = (ab - c²) - (ca - b²)
=> (ab - c²) + (bc - a²) = 2(ca - b²)
=> (ab - c² + bc - a²)
=> b(a + c) - (a² + c²)
=> b(2b) - [ a² + c² + 2ac - 2ac ]---from--(1)
=> 2b² - [ (a + c)² - 2ac ]
=> 2b² - [ (2b)² - 2ac] ----from--(1)
=> 2b² - 4b² + 2ac
=> -2b² + 2ac
=> 2(ac - b²)
R.H.S = L.H.S
Given that:-
a,b,c are in A.P
D = b-a = c-b
=> 2b = a+c-----------(1)
In A.P
D = (ca - b²) - (bc - a²) = (ab - c²) - (ca - b²)
=> (ab - c²) + (bc - a²) = 2(ca - b²)
=> (ab - c² + bc - a²)
=> b(a + c) - (a² + c²)
=> b(2b) - [ a² + c² + 2ac - 2ac ]---from--(1)
=> 2b² - [ (a + c)² - 2ac ]
=> 2b² - [ (2b)² - 2ac] ----from--(1)
=> 2b² - 4b² + 2ac
=> -2b² + 2ac
=> 2(ac - b²)
R.H.S = L.H.S
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