<s>[INST] Prove that $ a, b, c $ are in arithmetic sequence for the given condition. A question on my mathematics textbook, Mathematics—Textbook for Class XI, goes thus:

If $a\left(\frac{1}{b} + \frac{1}{c}\right), b \left(\frac{1}{c} + \frac{1}{a}\right)$ and $c \left(\frac{1}{a} + \frac{1}{b}\right)$ are in arithmetic sequence, prove that $a, b, c$ are also in arithmetic sequence.

I've tried solving the question by proceeding with this as the first step:
\begin{equation}
b \left(\frac{1}{c} + \frac{1}{a}\right) - a \left(\frac{1}{b} + \frac{1}{c}\right) = c \left(\frac{1}{a} + \frac{1}{b}\right) - b \left(\frac{1}{c} + \frac{1}{a}\right)
\end{equation}
However, I've been unable to reach this step:
\begin{equation}
b - a = c - b
\end{equation}
which, I believe, is required to arrive at to prove that $a, b, c$ are indeed in arithmetic sequence.
A guidance towards the right steps would be helpful.
 [/INST] Hint: Let $x_1=a\bigg(\frac{1}{b}+\frac{1}{c}\bigg)$, $x_2=b\bigg(\frac{1}{a}+\frac{1}{c}\bigg)$  and
$x_3=c\bigg(\frac{1}{a}+\frac{1}{b}\bigg)$.
Then
$$
x_3-2x_2+x_1=\frac{(ab+ac+bc)(a-2b+c)}{abc}
$$
Can you finish from here ?
</s>