What is a function $f(n)$ such that $t_n \in \Theta(f(n))$

Question

Let $t_n$ be defined as follows:

$$ t_n = \frac{571}{98}*7^n - \frac{45}{14}*5^n +\frac{6}{7}*n*5^n $$

I am looking for a function $f(n)$ such that $t_n \in \Theta(f(n))$.

I am confused at how I can find the $\Theta$

Can you please guide me?

Thanks

Answer 1

Let $A, B, C$ be positive constants. Then since: $$ \lim_{n \to \infty} \frac{A \cdot 7^n - B \cdot 5^n + C \cdot n 5^n}{7^n} = A $$

it follows that $t_n \in \Theta(7^n)$.