If a= b+c/2, c=b+a/2 and b is mean proportional between a& c, then prove 1/a + 1/c = 2/b.

Expert Answers

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Hi, dilipk,

First, by mean proportional, I believe your teacher means that b would be the geometric mean of a and c.  So:

b = sqrt( ac )

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

What I would do first is square each side.  Then, we have:

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

Foiling the right side:

b^2 = b^2 + ba/2 + bc/2 + ca/4

The b^2 terms would cancel out.  So:

0 = ba/2 + bc/2 + ca/4

Multiplying each side by 4

0 = 2ba + 2bc + ca

Subtracting the ca over:

-ca = 2ba + 2bc

Dividing each side by ca:

-1 = 2b/c + 2b/a

Dividing each side by 2b:

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

Rewritten:

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

Now, the left side matches what you have, but not the right side.  I suspect something went wrong with the solution you have.

Good luck, dilipk.  I hope this will help.

Till Then,

Steve

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