Question

if 1/a,1/b,1/c are in a.p show that a,b,c are in ap

Answer 1

As 1/a+b , 1/2b and 1/b+c are in an AP, there is a common difference.

∴1/2b−1/a+b=1/b+c−1/2b

1/b=1/b+c+1/a+b

1/b=a+b+b+c/(a+b)(a+b)

1/b=a+2b+c/(a+b)(a+b)

1/b=a+2b+c/ab+ac+bc+b2

1=ab+2b

+b/cab+ac+bc+

${b}^{2}$

ab+ac+bc+b2=ab+2

${b}^{2}$

+bc

ac+b2=2

${b}^{2}$

ac=

${b}^{2}$

ba=cb

Therefore a,b,c have a common ratio and are in a GP.