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### Solving Recurrence equation for $n$

I have a recurrence equation $t_n$ which is defined as

$$t_n=2.5t_{n-1}-1.5t_{n-4},\qquad \text{for }n\geq5$$

Where \begin{align} t_1&=5\ t_2&=10.5\ t_3&=26.25\ t_4&=62.625 \end{align}

I need to solve my recurrence equation for $n$ when $t_n=13\times10^{10}$. How can I achieve this in Sage?

### Solving Recurrence equation for $n$

I have a recurrence equation $t_n$ which is defined as

$$t_n=2.5t_{n-1}-1.5t_{n-4},\qquad \text{for }n\geq5$$

Where \begin{align} t_1&=5\ t_2&=10.5\ t_3&=26.25\ t_4&=62.625 \end{align}Where $t_1=5$, $t_2=10.5$, $t_3=26.25$, and $t_4=62.625$.

I need to solve my recurrence equation for $n$ when $t_n=13\times10^{10}$. How can I achieve this in Sage?