Question

The following geometric sequences represent the populations of two bacterial cultures at the 1-hour mark, the 2-hour mark, the 3-hour mark, and so on. Culture A starts with more bacteria, but culture B has a ratio of increase that is larger. Which culture will have the greatest population at the 19-hour mark?

A. 800, 1,200, 1,800, 2,700, ...

B. 5, 10, 20, 40, ...

Answer 1

The explicit formula for geometric sequence is given by:
a(n) = a(1)r^(n-1)

Where,
a(n) = population after time n
a(1) = population at the start
r = common ratio

For bacteria A;
a(1) = 800
r = 120/800 = 1800/1200 = 2700/1800 = 1.5
Then, after 19 hours;
a(19) = 800*1.5^(19-1) = 1182313.504 ≈ 1182314

Fro bacteria B;
a(1) = 5
r = 10/5 = 20/10 = 40/20 = 2
Then, after 19 hours;
a(19) = 5*2^(19-1) = 1310720

Therefore, culture B will have a greater population after 19-hour mark.