Let's analyze each sequence given in the problem to determine if it is a geometric sequence.

A geometric sequence has the form $U_n = U_1 \cdot q^{n-1}$, or equivalently, the ratio between consecutive terms $\frac{U_{n+1}}{U_n} = q$ is constant.

Sequence Analysis

a) $U_n = 6n$

• Here, $U_n$ depends linearly on $n$. The ratio $\frac{U_{n+1}}{U_n} = \frac{6(n+1)}{6n} = \frac{n+1}{n}$ is not constant.
• Conclusion: Not a geometric sequence.

b) $U_n = 2^{n-3} \cdot 3^{n+2}$

• Simplify the expression: $U_n = \frac{2^n}{2^3} \cdot 3^n \cdot 3^2 = 12 \cdot (2 \cdot 3)^n = 12 \cdot 6^n$.
• Clearly, this is in the form of a geometric sequence with $q = 6$ and first term $U_0 = \frac{12}{6^0} = 12$.
• Conclusion: Geometric sequence, $q = 6$, $U_0 = 12$.

c) $U_n = \frac{3^n}{5^{n+1}}$

• Rewrite: $U_n = \frac{3^n}{5 \cdot 5^n} = \frac{1}{5} \cdot \left(\frac{3}{5}\right)^n$.
• This is in the form of a geometric sequence with $q = \frac{3}{5}$ and $U_0 = \frac{1}{5}$.
• Conclusion: Geometric sequence, $q = \frac{3}{5}$, $U_0 = \frac{1}{5}$.

d) $U_n = -3 \cdot 10^n$

• Rewrite: $U_n = (-3) \cdot (10)^n$.
• This is in the form of a geometric sequence with $q = 10$ and $U_0 = -3$.
• Conclusion: Geometric sequence, $q = 10$, $U_0 = -3$.

e) Recursive sequence: