Nine-banded armadillos always give birth to quadruplets—four identical babies (called pups). Jenny’s armadillo has produced four female pups. Jenny realizes that each of those pups could eventually have four more armadillos. a. How many armadillos would Jenny have if the four pups grew up to have four more pups each?
b. What if every generation of Jenny’s armadillos was female? How many armadillos would there be if you only count the ones in the 5th generation? Can you write a formula for the number of armadillos in any given generation? Please explain what each variable and each number in your formula means.
c. According to this formula the world should be overrun with armadillos after 20 to 25 generations. Why isn’t this true in real life?

a. 16; b. aₙ = 4ⁿ⁻¹; c. predation and carrying capacity will limit the population

Step-by-step explanation:

a. The first female had 4 pups.

If each of them had 4 pups, there would be 4 × 4 = 4² = 16 pups.

b. 1st generation                        =     1 pup

2nd generation = 4 ×  1 pup      =    4 pups

3rd generation  = 4 ×  4 pups    =  16 pups

4th generation  = 4 × 16 pups    = 64 pups

5th generation  = 4 × 64 pups = 256 pups

The sequence is           1,   4, 16, 64, 256, …

We can also write it as 4⁰, 4¹, 4²,  4³,   4⁴, …

Note that the exponent is one less than the number of the generation.

Thus, the general formula for the nth term, aₙ, is

aₙ = 4ⁿ⁻¹, where

n = the number of the generation

a = the number of pups in that generation.

c. a₂₅ = 4²⁵⁻¹ = 4²⁴ ≈ 281 000 000 000 000 or 281 trillion

The population would not reach that number because there would not be enough food for all those pups. Pups would die of starvation until their numbers equalled the carrying capacity of the ecosystem and would level off at that point.

There would also be increased predation by coyotes, bobcats, etc. This would also keep the population growth in check,

expression #3 is not equivalent. the others are.