Arithmetic sequence and arithmetic mean are correlated like that $$ a_n=\frac{a_{n-1}+a_{n+1}}{2} $$ So all elements of arithmetic sequence $a_n=a+(n-1)r$ are satisfy that. $$ \frac{a+(n-2)r+a+nr}{2}=a+(n-1)r $$ The same with geometric sequence $$ a_n=\sqrt{a_{n-1}a_{n+1}} $$ for $a_n=aq^{n-1}$ we have $$ \sqrt{aq^{n-2}aq^n}=\sqrt{a^2q^{2(n-1)}}=aq^{n-1} $$ But what with harmonic sequence? $a_n=\frac{1}{a+(n-1)r}$ It doesn't satisfy harmonic mean. $$ \frac{2}{\frac{1}{a+(n-1)r}+\frac{1}{a+(n+1)r}}=\frac{\left(a+(n-1)r\right)\left(a+(n+1)r\right)}{a+nr}\not=\frac{1}{a+nr} $$
How looks like the sequence which satisfy harmonic sequence?
For arithmetic sequence I can start like that $$ a_n=\frac{a_{n-1}+a_{n+1}}{2}\Rightarrow 2a_n=a_{n-1}+a_{n+1}\Rightarrow a_n-a_{n-1}=a_{n+1}-a_n $$ So difference should be constant. We can start from any number $a$ and add some multiply of another number $r$. $a_n=a+nr$. For geometric mean: $$ a_n=\sqrt{a_{n-1}a_{n+1}}\Rightarrow a_n^2=a_{n-1}a_{n+1}\Rightarrow \frac{a_n}{a_{n-1}}=\frac{a_{n+1}}{a_n} $$ So quotient should be constant. $a_n=aq^n$. For harmonic mean $$ a_n=\frac{2}{\frac{1}{a_{n-1}}+\frac{1}{a_{n-1}}}\Rightarrow a_n=\frac{2a_{n-1}a_{n+1}}{a_{n-1}+a_{n+1}} $$ It isn't so nice. I found this form (maybe is useful) $$ \frac{a_{n+1}}{a_{n-1}}=\frac{a_{n+1}-a_n}{a_n-a_{n-1}} $$ What kind of method should I use to find this sequence?
Instead of just defining $a_n$ in terms of $a_{n+1}$ and $a_{n-1}$ and trying to expand the $a_{n+1},a_{n-1}$ function, I would recommend trying to isolate $a_{n+1}$
Arithmetic:
$a_n = \frac{a_{n+1}+a_{n-1}}{2}$
$2*a_n = a_{n+1}+a_{n-1}$
$a_{n+1} = 2 * a_n - a_{n-1} = a_n + (a_n - a_{n-1})$
Geometric:
$a_n = √(a_{n+1}a_{n-1})$
$(a_n)^2 = a_{n+1}a_{n-1}$
$a_{n+1} = \frac{(a_n)^2}{a_{n-1}} = a_n * \frac{a_n}{a_{n-1}}$
Harmonic:
$a_n = \frac{2}{\frac{1}{a_{n+1}}+\frac{1}{a_{n-1}}}$
$\frac{2}{a_n} = \frac{1}{a_{n+1}}+\frac{1}{a_{n-1}}$
$\frac{1}{a_{n+1}}=\frac{2}{a_n}-\frac{1}{a_{a-1}} = \frac{1}{a_n}+(\frac{1}{a_n}-\frac{1}{a_{a-1}})$
Do you know what to do next?