﻿ If \dfrac{1}{a},\dfrac{1}{b}, \dfrac{1}{c} is an arithmetic sequence, and a\ne b , verify a, b, c

#### Question

If \dfrac{1}{a},\dfrac{1}{b}, \dfrac{1}{c} is an arithmetic sequence, and a\ne b , verify a, b, c could not form an arithmetic sequence.

Collected in the board: Arithmetic sequence

Steven Zheng posted 1 month ago

Since \dfrac{1}{a},\dfrac{1}{b}, \dfrac{1}{c} is an arithmetic sequence

Let u be the common difference of the sequence,

\dfrac{1}{a} = \dfrac{1}{b}-u
(1)
\dfrac{1}{c} = \dfrac{1}{b}+u
(2)

Get reciprocals of (1) and (2)

a = \dfrac{1}{ \dfrac{1}{b}-u} = \dfrac{b}{1-bu}

c = \dfrac{1}{\dfrac{1}{b}+u} = \dfrac{b}{1+bu}

b-a = b - \dfrac{b}{1-bu} = b(1-\dfrac{1}{1-bu}) =- \dfrac{b^2u}{1-bu}

c-b= \dfrac{b}{1+bu}-b = b(\dfrac{1}{1+bu}-1 )=-\dfrac{b^2u}{1+bu}

Therefore,

b-a \ne c-b, a,b,c is not an arithmetic sequence.

Similarly a, c, b is also not an arithmetic sequence.

a, b, c could not form an arithmetic sequence in any order.

Steven Zheng posted 1 month ago

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