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if a,b,c are in ap , prove that a(b+c)/bc, b(c+a)/ca, c(a+b)/ab are also in a.p.

if a,b,c are in ap , prove that a(b+c)/bc, b(c+a)/ca, c(a+b)/ab are also in a.p.

Grade:12th pass

1 Answers

Shashank Singh
askIITians Faculty 57 Points
5 years ago
a,b,c are in ap
b= (a+c)/2
now for a(b+c)/bc, b(c+a)/ca and c(a+b)/ab
(ab+ac)/bc + (ca+cb)/ab
= a/c+a/b+c/b+c/a
=a/c+c/a+(a+c)/b= a/c+c/a+2
= (v(c/a)+v(a/c))^2= (c+a)^2/ac= (c+a)(c+a)/ac
= 2b(a+c)/ac
therefore we can say that a(b+c)/bc, b(c+a)/ac and c(a+b)/ab are in AP....... [Ans]

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