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

Question

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

Uniteasy Admin · Accepted Answer

Since $$\dfrac{1}{a},\dfrac{1}{b}, \dfrac{1}{c}$$ is an arithmetic sequence
Let $$u$$ be the common difference of the sequence, 
$$\dfrac{1}{a} = \dfrac{1}{b}-u$$n
$$\dfrac{1}{c} = \dfrac{1}{b}+u $$n
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, 
$$b-a 
e c-b$$, $$a,b,c$$ 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.

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