. If (a -- c) is the geometric mean of (a + 2b + c) and (a – 2b + c), show that a, b, c are in
continued proportion.
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Please refer the attachment:)
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This gives (a + 2b + c)/(a - c) = (a - c)/(a - 2b + c)
=> (a + 2b + c)(a - 2b + c) = (a - c)^2
=> (a + c)^2 - 4b^2 = (a - c)^2
=> a^2 + c^2 + 2ac - 4b^2 = a^2 + b^2 - 2ac
=> 4ac = 4b^2
=> ac = b^2
=> b/a = c/b
This proves that b is the mean proportion between a and c or b/a = c/b.
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