Question Find the 10th term of the geometric sequence \( 8,-24,72, \ldots \) Answer Attempt 1 out of 3 Subdeo Submit Answer

To find the 10th term of the geometric sequence, we first need to identify the common ratio \( r \). In this sequence, you can find \( r \) by dividing the second term by the first term: \[ r = \frac{-24}{8} = -3. \] Now that we have \( r = -3 \), we can use the formula for the nth term of a geometric sequence, which is given by: \[ a_n = a_1 \cdot r^{(n-1)}, \] where \( a_1 \) is the first term. Plugging in the values, we have: \[ a_{10} = 8 \cdot (-3)^{(10-1)} = 8 \cdot (-3)^9. \] Calculating \( (-3)^9 = -19683 \), we find: \[ a_{10} = 8 \cdot (-19683) = -157464. \] So, the 10th term is \(-157464\). ----- In the realm of geometric sequences, the idea of exponential growth or decay makes it a riveting subject! Each term is formed by multiplying the previous term by a consistent number, which can lead to some truly astounding results. Think about how fast these numbers can grow or shrink—like the way interest compounds in finance! Another way to visualize geometric sequences is through their graphical representation. Plotting the terms of the sequence can create fascinating patterns, often revealing their behavior over time. Whether upward or downward, each step hints at the underlying consistency of our journey through numbers—quite thrilling, right?