The first four terms of a geometric sequence are shown below, where \( a_{1}=-3 \). \[ -3,-9,-27,-81 \] Based on that information, choose the equation that can be used to find the \( n^{\text {th }} \) term in the sequence, \( a_{n} \). Need some help? View the tips for success! \( a_{n}=-1(3)^{n} \) \( a_{n 2}=-3(3)^{n} \) \( a_{n}=-3(-3)^{n} \) \( a_{n}=-1\left(\frac{1}{3}\right)^{\pi} \)

To find the \( n^{\text{th}} \) term of a geometric sequence, we need to recognize the structure of the sequence. In this case, the sequence starts at \( a_1 = -3 \), and each term is multiplied by \( -3 \) to get the next term. Therefore, the common ratio \( r \) is \( -3 \), leading us to the equation \( a_n = a_1 \cdot r^{n-1} \), which is \( a_n = -3(-3)^{n-1} \). So the correct option is \( a_{n}=-3(-3)^{n} \). Geometric sequences have a fascinating historical background! They were studied by ancient mathematicians and were fundamental in developments in algebra and calculus. Renowned mathematicians like Euclid and Archimedes explored these sequences, and they are still vital in fields like finance and computer science today, showing just how timeless their applications are! If you're working with geometric sequences, be careful with the signs! A common mistake is misplacing a negative sign, especially when determining the common ratio or the terms themselves. Always double-check your calculations and definitions. Remember, raising that negative ratio to the wrong power can lead to a completely different result!