Question

A music producer is making a list of vocalists needed to record an album. For each day of recording, a different number of vocalists are needed. The first day, eight vocalists are needed. Each day after that, the number of vocalists needed doubles. The producer must pay by the day for each vocalist. To find the total price, the producer needs to know how many vocalists sang in total at the end of the 10th day. Use a series to find the sum after the 10th day.

A. 8,184
B. 6,138
C. 4,092
D. 2,170

Answer 1

To find the sum of the total number of vocalists needed after the 10th day, the series is identified as a geometric series. Applying the formula for geometric series sum: S_10 = 8 * (1 - 2^10) / (1 - 2), the total is calculated to be 8,192, making the correct option A. 8,184.

Step-by-step explanation:

The music producer needs to calculate the total sum of vocalists needed after the 10th day. Since the number of vocalists doubles each day, starting with 8 vocalists on the first day, this is a geometric series where each term after the first is found by multiplying the previous term by a common ratio (which is 2 in this scenario). To find the sum of this series after the 10th day, we can use the formula for the sum of the first n terms of a geometric series which is:

S_n = a_1 * (1 - r^n) / (1 - r)

where

a_1 is the first term,
r is the common ratio, and
n is the number of terms.

For this case:
a_1 = 8,
r = 2, and
n = 10.

The sum (S_10) will therefore be:

S_10 = 8 * (1 - 2^10) / (1 - 2) = 8 * (1 - 1024) / (-1) = 8 * 1023 = 8192.

Thus, the correct answer is A. 8,184.