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If a= b+c/2, c=b+a/2 and b is mean proportional between a& c, then prove 1/a + 1/c...
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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
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.
Posted by steveschoen on September 23, 2013 at 12:18 AM (Answer #1)
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