Find the sum of the first 10 terms of the following sequence. Round to the nearest 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 determine the common ratio \( r \) of this geometric sequence. We can calculate \( r \) by dividing the second term by the first term: \[ r = \frac{-63}{21} = -3 \] Next, we'll use the first term \( a = 21 \) and the common ratio \( r = -3 \) to find the sum of the first 10 terms of the geometric series. The formula for the sum \( S_n \) of the first \( n \) terms of a geometric series is: \[ S_n = a \frac{1 - r^n}{1 - r} \] Plugging in the values: - \( a = 21 \) - \( r = -3 \) - \( n = 10 \) We need to calculate \( r^n \): \[ r^{10} = (-3)^{10} = 59049 \] Now substituting these values into the sum formula: \[ S_{10} = 21 \frac{1 - 59049}{1 - (-3)} \] \[ = 21 \frac{1 - 59049}{1 + 3} \] \[ = 21 \frac{1 - 59049}{4} \] \[ = 21 \frac{-59048}{4} \] \[ = 21 \cdot -14762 \] \[ = -310,002 \] Thus, the sum of the first 10 terms of the sequence is \[ \boxed{-310002}. \]