Question Find the 14 th term of the geometric sequence \( 5,-10,20, \ldots \) (©) Watch Video Answer Attempt 1 out of 3 Submit Answer

To find the 14th term of the geometric sequence \(5, -10, 20, \ldots\), we first identify the first term \(a = 5\) and the common ratio \(r\). The common ratio can be calculated as \(r = \frac{-10}{5} = -2\). The formula for the \(n\)-th term of a geometric sequence is given by \(a_n = a \cdot r^{n-1}\). So for the 14th term: \[ a_{14} = 5 \cdot (-2)^{14-1} = 5 \cdot (-2)^{13} \] Calculating \( (-2)^{13} = -8192 \), we get \( a_{14} = 5 \cdot (-8192) = -40960\). Therefore, the 14th term is \(-40960\). Did you know that sequences can be found in various natural patterns? For instance, geometric sequences often appear in nature, like the branching of trees or the arrangement of leaves around the stem. This fascinating connection between math and nature shows how deeply intertwined mathematical concepts are with the world around us! Also, when working with geometric sequences, it’s important to double-check your calculations, especially with powers and signs. Mistakes often come from miscalculating the common ratio or the exponent during computations. Take a moment to verify each step; it'll save you from those pesky errors and lead to success in solving more complex problems!