Finding the nth term of a geometric sequence

In a geometric sequence, each term is found by multiplying the previous term by a constant ratio. Let's check the ratio for the given sequence: Given the sequence: \[ -4, -16, -64, -256, -1024, -4096 \] To find the common ratio (\( r \)), divide the second term by the first term: \[ r = \frac{-16}{-4} = 4 \] To confirm this, let's check the ratio between subsequent terms: \[ \frac{-64}{-16} = 4 \] \[ \frac{-256}{-64} = 4 \] \[ \frac{-1024}{-256} = 4 \] \[ \frac{-4096}{-1024} = 4 \] It is consistent, so the common ratio is \( 4 \). The formula for the nth term (\( a_n \)) in a geometric sequence is: \[ a_n = a_1 \cdot r^{n-1} \] Where: - \( a_1 \) is the first term - \( r \) is the common ratio - \( n \) is the term number Given \( a_1 = -4 \) and \( r = 4 \): \[ a_n = -4 \cdot 4^{n-1} \] Let's review the given answer choices: A. \( a_n = -(4)^n \) B. \( a_n = -4n \) C. \( a_n = (-4)^n \) D. \( a_n = -n^2 \) Among these choices, choice A can be rewritten and simplified to match our sequence's formula: \[ a_n = -(4^n) \] This is the same as: \[ a_n = -4 \cdot 4^{n-1} \] Because, \[ -(4)^n = -4 \cdot 4^{n-1} \] Thus, the correct equation to find the nth term in the sequence is: **A. \( a_n = -(4)^n \)**