Question

The sequence below shows the number of bacteria Arjun observed each hour for a science experiment:

5, 20, 80, 320, 1,280,...

Which recursive function describes the number of bacteria observed at the nth hour?
a) f(1) = 5; f(n + 1) = 4f(n)
b) f(1) = 1; f(n + 1) = 4f(n)
c) f(1) = 1; f(n + 1) = 5f(n)
d) f(1) = 5; f(n + 1) = 2f(n) + 10

Answer 1

The correct recursive function that describes the bacterial growth at the nth hour is f(1) = 5; f(n + 1) = 4f(n), which is option (a). This shows the bacteria count is quadrupling every hour, which is consistent with exponential growth.

Step-by-step explanation:

The student is asking about a recursive function that describes the number of bacteria observed at the nth hour, given a sequence of numbers representing bacterial counts. After observing the given sequence, we can see that the number of bacteria is increasing each hour by a factor of 4 (20 is 4 times 5, 80 is 4 times 20, and so on). Therefore, the recursive function that correctly describes this pattern is:

f(1) = 5; f(n + 1) = 4f(n).

Breaking down this function:

• f(1) = 5 means that the initial number of bacteria (at the first hour) is 5.
• f(n + 1) = 4f(n) indicates that at each subsequent hour, the number of bacteria is 4 times the previous hour's bacteria count.

The corresponding correct answer to the student's question is option (a).