# 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 $t_1=5$, $t_2=10.5$, $t_3=26.25$, 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?