Find whether $\sum_{n=1}^\infty \frac{2^{3n-3}3^{n+2}}{5^{2n-2}} $ is geometric and convergent and if so, state the value of the sum $$\sum_{n=1}^\infty  \frac{2^{3n-3}3^{n+2}}{5^{2n-2}} $$
First I tried to find the ratio by doing $\frac{a_{n+1}}{a_n}$. Many calculations were made, and the result was $24/25$, so it is geometric. This is between -1 and 1, and therefore the series is convergent.
Then I did $$\sum_{d=1}^\infty  \frac{a_1}{1-24/25} = (...) = 675$$
Is this correct?

$$\sum_{n=1}^\infty  \frac{2^{3n-3}3^{n+2}}{5^{2n-2}} =\frac{9.25}{8}\sum_{n=1}^{\infty } \frac{8^n.3^n}{25^n}=\frac{225}{8}\frac{\frac{24}{25}}{1-\frac{24}{25}}=675$$
So your answer is true.
