Find the sum of the first 10 terms of the following sequence. Round to the rearest hundredth if necessary. \[ 21, \quad-63, \quad 189, \ldots \]

To find the sum of the first 10 terms of the sequence \( 21, -63, 189, \ldots \), we first need to recognize that this sequence is a geometric series. The first term \( a \) is \( 21 \), and to find the common ratio \( r \), we can divide the second term by the first term: \[ r = \frac{-63}{21} = -3. \] Now, we can use the formula for the sum of the first \( n \) terms of a geometric series: \[ S_n = a \frac{1 - r^n}{1 - r}, \] where \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. For our problem, we are seeking \( S_{10} \): \[ S_{10} = 21 \frac{1 - (-3)^{10}}{1 - (-3)}. \] Calculating \( (-3)^{10} \): \[ (-3)^{10} = 59049, \] Now substituting it back into the sum formula: \[ S_{10} = 21 \frac{1 - 59049}{1 + 3} = 21 \frac{1 - 59049}{4} = 21 \frac{-59048}{4}. \] Calculating further: \[ S_{10} = 21 \cdot (-14762) = -308982. \] Thus, the sum of the first 10 terms of the sequence is: \[ \boxed{-308982}. \]