Question

4. A colony of bacteria is growing in such a way that each generation is 2.5 times as large as the previous generation. Suppose there are 450 bacteria in the first generation.

a) Show that nº generation is 180(2.5)^n

b) How many will there be in the fifth generation?

c) What will be the total number of bacteria in the first tenth generations to the nearest integer? [10 marks]
​

Answer 1

Explanation:

I think there are some typos in a)

anyway, this is a geometric sequence.

that means that after a starting value each new term is created by multiplying the previous term by a certain factor.

a)

s1 = 450

factor f = 2.5

so,

s2 = 450×2.5 = 1125

and so on.

from that we see that

sn = 450×(2.5)^(n-1)

and as

450/2.5 = 180

or

450 = 180×2.5

we can "extract" another 2.5 factor from the basic 450.

so,

sn = 180×2.5×(2.5)^(n-1) = 180×(2.5)^n

b)

s5 = 180×(2.5)⁵ = 17,578.125 or rounded 17578

c)

the question is unclear. are the bacteria of previous generations still there, when a new one is generated ?

then the 2.5 factor creates the sum implicitly, and the total number of bacteria in the first 10 generations is simply the 10th term s10.

s10 = 180×(2.5)¹⁰ = 1716613.76953125 = 1,716,614

or is it meant to sum the numbers of each generation (the old bacteria "go away", when the next generation is generated) ?

the sum of the first n terms of such a geometric sequence is

sum = a1×(1 - f^n)/(1 - f)

so, for our first ten terms

sum = 450×(1 - 2.5^10)/(1 - 2.5) =

= 450×-9535.7431640625/-1.5 =

2860722.94921875 = rounded 2,860,723