\( \sum _ { r = 1 } ^ { m } 5 ^ { r } = 19530 \)

To solve \( \sum _ { r = 1 } ^ { m } 5 ^ { r } = 19530 \), we note that this is a geometric series. The formula for the sum of a geometric series is \( S_n = a \frac{(1 - r^n)}{1 - r} \), where \( a \) is the first term (5 in this case), \( r \) is the common ratio (also 5), and \( n \) is the number of terms. For our sum, we rearrange: \[ S_m = 5 \frac{(1 - 5^m)}{1 - 5} = 5 \frac{(1 - 5^m)}{-4} \] This simplifies to: \[ S_m = -\frac{5}{4} (1 - 5^m) = \frac{5(5^m - 1)}{4} \] Setting this equal to 19530: \[ \frac{5(5^m - 1)}{4} = 19530 \] Multiplying both sides by 4 gives: \[ 5(5^m - 1) = 78120 \] Dividing both sides by 5: \[ 5^m - 1 = 15624 \] Adding 1: \[ 5^m = 15625 \] Since \( 5^6 = 15625 \), it follows that \( m = 6 \). Thus, \( m = 6 \).