To find $ n $ we use formula

$$ \color{blue}{S_n = a_1 \cdot \frac{1-r^n}{1-r}}$$

In this example we have $ a_1 = 10 ~~,~~ r = \frac{ 61 }{ 50 } ~~,~~ S_n = 360 $. After substituting these values to above formula, we obtain:

$$ \begin{aligned} S_n &= a_1 \cdot \frac{1-r^n}{1-r} \\[1 em] 360 &= 10 \cdot \frac{ 1-\left( \frac{ 61 }{ 50 } \right)^n}{1 - \left( \frac{ 61 }{ 50 } \right)} \\[1 em] 1-\left( \frac{ 61 }{ 50 } \right)^n &= \frac{ 360}{ 10} \cdot \left(1 - \left( \frac{ 61 }{ 50 } \right) \right) \\[1 em] 1-\left( \frac{ 61 }{ 50 } \right)^n &= -\frac{ 198 }{ 25 } \\[1 em] \left( \frac{ 61 }{ 50 } \right)^n &= \frac{ 223 }{ 25 } \\[1 em] \log \left( \left( \frac{ 61 }{ 50 } \right)^n \right) &= \log \left(\frac{ 223 }{ 25 } \right) \\[1 em] n \cdot \log \left( \left( \frac{ 61 }{ 50 } \right) \right) &= \log \left(\frac{ 223 }{ 25 } \right) \\[1 em] n &\approx 11.00471 \end{aligned} $$

Since $ n $ is not a positive integer we conclude that the problem has no solution.

Geometric sequences