3. An exponential sequence is defined by \( 30, 15, 7 \frac{1}{2}, \ldots \). Find the sum to infinity of this sequence. A. 15 B. 30 C. 45 D. 60

To find the sum to infinity of the given exponential sequence, we first need to determine the common ratio. The common ratio \( r \) can be found by dividing the second term by the first term: \[ r = \frac{15}{30} = \frac{1}{2} \] Now, we confirm the common ratio by checking the next pair of terms: \[ \frac{7 \frac{1}{2}}{15} = \frac{7.5}{15} = \frac{1}{2} \] This confirms that \( r = \frac{1}{2} \). The formula for the sum to infinity \( S \) of a geometric series is given by: \[ S = \frac{a}{1 - r} \] where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = 30 \) and \( r = \frac{1}{2} \). Plugging in these values: \[ S = \frac{30}{1 - \frac{1}{2}} = \frac{30}{\frac{1}{2}} = 30 \times 2 = 60 \] So the sum to infinity of the sequence is \( \boxed{60} \).