Find the 12th term of the geometric sequence. 1,-1,1,ldots Show your work here

To find the 12th term of the geometric sequence $1, -1, 1, \ldots$, we need to identify the common ratio between consecutive terms.<br /><br />The common ratio (r) can be found by dividing any term by its previous term. Let's calculate the common ratio using the first two terms:<br /><br />$r = \frac{-1}{1} = -1$<br /><br />Now that we have the common ratio, we can use the formula for the nth term of a geometric sequence:<br /><br />$a_n = a_1 \cdot r^{(n-1)}$<br /><br />where $a_n$ is the nth term, $a_1$ is the first term, and n is the term number.<br /><br />In this case, $a_1 = 1$, $r = -1$, and $n = 12$. Plugging these values into the formula, we get:<br /><br />$a_{12} = 1 \cdot (-1)^{(12-1)}$<br /><br />$a_{12} = 1 \cdot (-1)^{11}$<br /><br />Since any number raised to an odd power is negative, we have:<br /><br />$a_{12} = -1$<br /><br />Therefore, the 12th term of the geometric sequence $1, -1, 1, \ldots$ is $-1$.