In the geometric sequence with a first term of $6$ and a second term of $-6$, what is the $205^{th}$ term?
To find the $205^{th}$ term of a geometric sequence, I need to know the common ratio between consecutive terms and the first term. I already have the first term, which is $6$. To find the common ratio, I can divide the second term by the first term. So, the common ratio is $-6 \div 6 = -1$. Now, I can use the formula for the $n^{th}$ term of a geometric sequence, which is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number. Plugging in the values I have, I get $a_{205} = 6 \cdot (-1)^{204}$. I can simplify this by noting that $(-1)^{204}$ is just $1$, since $(-1)$ raised to an even power is always $1$. So, the $205^{th}$ term is $6 \cdot 1 = 6$. 