Final answer:
The number of students that have heard the rumor on day 6 is 256, found by identifying the common ratio in the geometric sequence and then calculating the 6th term. Therefore correct option is A
Step-by-step explanation:
To solve the problem of calculating the number of students that have heard a rumor on day 6 given a geometric sequence, we first recognize that we have information about days 3 and 8. We can use the formula for the nth term of a geometric sequence, which is a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.
First, we find the common ratio using the information given for days 3 and 8:
a_3 = 32, a_8 = 1024
Because the days are equidistant, we can divide the term of day 8 by the term of day 3 to find r^5 (since 8 - 3 = 5):
1024 / 32 = r^5
32 = r^5
r = 2
Now, to find the number of students that have heard the rumor on day 6, we calculate the 6th term of the sequence:
a_6 = a_1 * r^(6-1)
We don't have a_1, but we can use a_3 to express a_6:
a_6 = (a_3 / r^(3-1)) * r^(6-1)
a_6 = (32 / r^2) * r^5
a_6 = (32 / 2^2) * 2^5
a_6 = (32 / 4) * 32
a_6 = 8 * 32
a_6 = 256
The number of students that have heard the rumor on day 6 is 256 students, which corresponds with option (A).