Please help us to find the general solution of this recurrence: $x_{n+1}=5^{n-1} x_n+3^n$ Please help us to find the general solution of this recurrence:
$$x_{n+1}=5^{n-1} x_n+3^n.$$
We found the solution for the associated homogeneous recurrence
$x_{n+1}=5^{n-1} x_n$
which is 
$$x_n =  5^{(n-1)(n-2)/2}x_1,$$
and we tried to use it to find a particular solution of the nonhomogeneous recurrence, but unfortunately we were not able to find it.

This is the route I usually take but....
$$\begin{align}
x_{n+1}&=5^{n-1} x_n+3^n \\
&=5^{n-1}(5^{n-2} x_{n-1}+3^{n-1})+3^n \\
&\quad = 5^{2n-(1+2)}x_{n-1}+(3 \cdot 5)^{n-1} +3^n \\
&=5^{2n-(1+2)}(5^{n-3}x_{n-2}+3^{n-2})+(3\cdot5)^{n-1}+3^n \\
&\quad =5^{3n-(1+2+3)}x_{n-2}+5^{2n-(1+2)}3^{n-2}+5^{n-1}3^{n-1}+3^n \\
&\vdots \\
&\vdots \quad \text{seeing a pattern...after i steps} \\
&\vdots \\
&=5^{(i+1)n+\frac{(i+1)(i+2)}{2}}x_{n-i}+\sum_{k=0} ^{i} 5^{kn-\frac{k(k+1)}{2}}3^{n-k} \\
&\vdots \\
&\vdots \quad \text{supposing that you do it n times, and set i=n...} \\
\\
&\vdots \\
&=5^{\frac{(n+1)(3n+2)}{2}}x_0+\sum_{k=0} ^{n} 5^{kn-\frac{k(k+1)}{2}}3^{n-k} \\
\end{align}$$
I'm not sure if this is any help to you but I couldn't resist taking a stab.
