Question

The number of students in a school who have heard a rumor at time, t, is modeled by a geometric sequence,

where the rumor was started on first day of school (day 1). If there were 32 students that have heard the rumor
on day 3, and 1,024 students that have heard the rumor on day 8, then what is the number of students that have
heard the rumor on day 6?
(A) 256
(B) 512
(C) 64
(D) 128

Answer 1

Day 3's 32 students multiplied by the common ratio (32) cubed yields 512 students on day 6, so option (B) is correct.

Here's how we can find the number of students who have heard the rumor on day 6:

Recognize the geometric sequence: We are given that the number of students follows a geometric sequence. This means that the number of students who have heard the rumor is multiplied by a constant factor (common ratio) each day.

Find the common ratio: We know that there were 32 students on day 3 and 1,024 students on day 8. To find the common ratio, we can divide the number of students on day 8 by the number of students on day 3: 1,024 students / 32 students = 32.

Calculate the number of students on day 6: Now that we know the common ratio is 32, we can find the number of students who have heard the rumor on day 6. Since day 6 is 3 days after day 3, we need to multiply the number of students on day 3 by the common ratio raised to the power of 3: 32 students *
`32^3` = 512 students.

Therefore, on day 6, there are 512 students who have heard the rumor.

Note: Options (A), (C), and (D) can be eliminated by checking if they are obtained by multiplying 32 (the number of students on day 3) by the common ratio (32) raised to different powers.

The correct answer is (B) 512.