Question

The trees at a national park have been increasing in numbers. There were 1,000 trees in the first year that the park started tracking them. Since then, there have been as many new trees each year. Create the sigma notation showing the infinite growth of the trees and find the sum, if possible. Year New trees 1 1000 2 200 3 40

Answer 1

Answer:

Step-by-step explanation:

There were 1000 trees in the first year and every year new trees are getting added.

The sequence formed for the new trees every year is

Year                1           2           3

New trees   1000      200      40

We see a geometric sequence has been formed by the new trees added

Ratio of the second year and 1st year trees added = $\frac{200}{1000}=\frac{1}{5}$

Similarly ratio of trees added in 3rd year to 2nd year = $\frac{40}{200}=\frac{1}{5}$

So there is a common ratio of $\frac{1}{5}$

Explicit formula of a geometric sequence representing growth of the trees by

$T_{n}=a(r)^{n-1}$

where a = number of trees grown first year

r = common ratio

n = number of years

Explicit formula showing the growth of the trees using sigma notation will be

$\sum_{n=1}^{\infty}1000(\frac{1}{5})^{n-1}$

And Formula for number of trees every year will be

$\sum_{n=1}^{\infty}1000+1000(\frac{1}{5})^{n-1}$

$\sum_{n=1}^{\infty}1000[1+(\frac{1}{5})^{n-1}]$

Sum of the trees will be

$S=\frac{a}{1-r}$

        = $\frac{2000}{1-\frac{1}{5}}$

        = $\frac{2000}{\frac{4}{5} }$

        = $\frac{2000\times 5}{4}$

        = 2500