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if a,b,and c are in an A.P. then prove a(b+c)/bc, b(c+a)/ca,c(a+b)/ab are in an A.P.?

if a,b,and c are in an A.P. then prove a(b+c)/bc, b(c+a)/ca,c(a+b)/ab  are in an A.P.?

Grade:12th pass

1 Answers

Arun
25763 Points
3 years ago
Dear Manju
 
Let solve this question in reverse.
Suppose we have to prove a, b, c are in A.P.
Hence
(2bc +2ab)/ca = (ab +ac)/bc + (ac +bc)/ab
Make LCM as abc nd then cancel out the denominator.
 
2 b²c + 2 b²a = a²b +a²c + ac² + bc²
b²c - bc² + b²a - ac² = a²b - b²a + a²c -b²c 
bc(b-c) + a (b-c)(b+c) = ab (a-b) + c(a-b)(a+b)
(b-c) (ab + bc + ca) = (ab + bc+ ca) (a-b)
b-c = a -b
2 b = a+c
Hence proved that a,b,c are in A.P.
 
Regards
Arun (askIITians forum expert)

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