Prove in harmonic development
Write these formulas in components create and also make use of the adhering to procedures :

$C_{2} \to C_{2} 2C_{3}$

$R_{3} \to R_{3}  R_{2}$ and also $R_{2} \to R_{2}R_{1}$
When these 2 procedures are done, after that the resulting component needs to be :
$$\left\begin{array}{lll} 1 & 0 & a \\ 0 & b & ba \\ 0 & 2cb & cb \end{array}\right = b(cb)  (ba)(2cb) =0 $$
which will certainly offer $$ \frac{2}{b} = \frac{1}{a} + \frac{1}{c}$$
OK, Chandru1 addressed it, so I'm gon na clarify the means I would certainly have approached it. (Which is in fact the very same strategy approximately getting my formulas in different ways.)
1/ An uniform system of formulas has a non  absolutely no remedy if the component is absolutely no. Consequently consider this formula :
$$\left\begin{array}{ccc} 1 & 2a & a \\ 1 & 3b & b \\ 1 & 4c & c \end{array}\right = 0$$
2/ Since we intend to look for a harmonic development, we require to take into consideration the inverses of $a,b$ and also $c$. If among these coefficients is absolutely no yet not the others, there can not be a harmonic development, yet you can additionally show that in these instances, the system of formulas can not have a non  absolutely no remedy, so they are dismissed anyhow. If 2 or even more of these coefficients are absolutely no, the system is underdetermined and also there is an infinity of remedies. So, it appears that there is a tiny technicality below in the inquiry.
Currently, thinking that all coefficients are various from absolutely no, we can revise the component as
$$\left\begin{array}{ccc} 1/a & 2 & 1 \\ 1/b & 3 & 1 \\ 1/c & 4 & 1 \end{array}\right = 0$$
Subtracting the 2nd row from the first and also 3rd row, you get
$$\left\begin{array}{ccc} 1/a1/b & 1 & 0 \\ 1/b & 3 & 1 \\ 1/c1/b & 1 & 0 \end{array}\right = 0$$
And this can be created relative to the 3rd column to get as an outcome :
$$\frac{1}{a}+\frac{1}{c}=\frac{2}{b}$$
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