Arithmetic and geometric sequences

Question

A microbiologist was studying the growth patterns of a bacterial culture. After twelve hours he counted 10 colonies in the petri dish. He noted that every 12 hours the number of colonies doubled. What type of sequence is being demonstrated?

Answer 1

Hi, Chloe.

You need to find out if this sequence is arithmetic or geometric. Let's figure out the sequence. If, after 12 hours, the count was 10, and that it had doubled from 12 hours prior, then the initial count was 5. Also, 12 hours after the count was 10, it would then be 20, and then 40, etc....

So the series would be :

5, 10, 20, 40, 80,.............

So we can find the pattern for the series:

5 x 2⁰ = 5

5 x 2¹ = 10

5 x 2² = 20

5 x 2³ = 40

.............................

Therefore we see that this is a geometric sequence in the form

a_n = a₁ (r^n-1)

Where a_{1 = 5} and r = 2

Therefore the sequence is:

a_n = 5(2^n-1)

Therefore,

a₁ = 5(1) = 5

a₂ = 5(2) = 10

a₃ = 5(4) = 20

a₄ = 5(8) = 40

And, indeed, this is a geometric sequence.

Answer 2

I believe the term you are looking for is exponential growth. You can usually find the doubling time by using what is called the rule of 70