Scientists have discovered a vaccine for a debilitating disease. This year there were 570,000 reported new cases of the disease in the United States. Research indicates, that with the new vaccination, new cases will decrease by two-thirds each year.

How many new cases do you estimate in 6 years?

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RenatoMattice RenatoMattice · Answer 1 · 2018-11-01T07:55:48+01:00

Answer: There are approximately 853827 new cases in 6 years.

Step-by-step explanation:

Since we have given that

Initial population = 570000

Rate at which population decreases is given by

[tex]\frac{2}{3}[/tex]

Now,

First year =570000

Second year is given by

[tex]570000\times (\frac{1}{3})[/tex]

Third year is given by

[tex]570000(\frac{1}{3})^2[/tex]

so, there is common ratio ,

it becomes geometric progression, as there is exponential decline.

so,

[tex]570000,570000\times \frac{1}{3},570000\times( \frac{1}{3})^2,......,570000\times (\frac{1}{3})^6[/tex]

a=570000

common ratio is given by

[tex]r=\frac{a_2}{a_1}=\frac{1}{3}[/tex]

number of terms = 6

Sum of terms will be given by

[tex]S_n=\frac{a(1-r^n)}{(1-r)}[/tex]

We'll put this value in this formula,

[tex]S_6=\frac{570000(1-(\frac{1}{3})^6}{(1-\frac{1}{3})}\\\\=853827.16[/tex]

So, there are approximately 853827 new cases in 6 years.