Question © Video Examples Write an explicit formula for \( a_{n} \), the \( n^{\text {th }} \) term of the sequence \( 8,-32,128, \ldots \)

To find the \( n^{th} \) term of the sequence \( 8, -32, 128, \ldots \), we first notice that this is a geometric sequence. The first term \( a_1 \) is 8, and the common ratio \( r \) can be calculated as follows: \( r = \frac{-32}{8} = -4 \). This means each term is multiplied by \(-4\) to get to the next. The explicit formula for a geometric sequence is \( a_n = a_1 \cdot r^{n-1} \). Plugging in our values, we get \( a_n = 8 \cdot (-4)^{n-1} \). As fantastical as it sounds, this sequence could represent growth trends in some sci-fi stories. Imagine a scenario where an intergalactic plant grows rapidly, reversing growth at certain intervals! This would result in fluctuations similar to negative energy states. Just think about all the wild stories that could unfold from a mysterious plant that grows and shrinks exponentially!