In an experiment, a certain colony of bacteria initially has a population of 50,000. A reading is taken every 2 hours, and at the end of every 2-hour interval, there are 3 times as many bacteria as before. a. Write a recursive definition for A(n), the number of bacteria present at the beginning of the nth time period. b. At the beginning of which interval are there 1,350,000 bacteria present?

1) Recursive definition:

2) At the beginning of the 4th interval

Explanation:

1)

The initial population of the bacteria at time zero is

Here we are told that the reading is taken every two hours; we call this time interval "n", so

And also, after every time interval n, the number of bacteria has tripled.

This means that when n = 1,

And when n=2,

Applied recursively, we get

And substituting p0,

(1)

2)

Here we want to find at the beginning of which interval there are

bacteria.

This means that we can rewrite eq.(1) as

By simplifying,

Which means that

However, this means that the number of bacteria is 1,350,000 after 3 time intervals; therefore, at the beginning of the 4th interval.

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