Question

1. a. Suppose the number of bacteria in a colony quadruples every hour.  Set up a recurrence relation for the number of bacteria in the colony at the end of n hours.

b. Find an explicit formula for the number of bacteria remaining in the colony after n hours.

c.  If 80 bacteria form a new colony, how many will be in the colony after three hours?

Answer 1

1) (a) Let us assume that the number of bacteria at present is = x.

As the number of bacteria in a colony quadruples every hour.

So, the number of bacteria after 1 hour = 4x

The number of bacteria after 2 hours = (4*4)x = 16x = 4²x

The number of bacteria after 2 hours = (4*4*4)x = 64x = 4³x

Similarly, the number of bacteria after n hours = 4ⁿx

1) (b) The explicit formula for the number of bacteria remaining in the colony after n hours is = 4ⁿx

Where, n = number of hours

x = number of bacteria in the beginning.

1) (c) Number of bacteria in the new colony = 80

So, the population of bacteria after three hours = 4³*80 = 64*80 = 5120 [As n = 3 and x = 80]

Hope this helps.