Question

The population of a type of local frog can be found using an infinite geometric series where a1 = 84 and the common ratio is one fifth. Find the sum of this infinite series that will be the upper limit of this population.

Answer 1

The series is 84(1+1/5+1/25+...)=84(1/(1-1/5)=84÷4/5=84×5/4=21×5=105. Upper limit is 105.

Answer 2

Answer:

The sum is 105

Step-by-step explanation:

Given that the population of a type of local frog can be found using an infinite geometric series where a1 = 84 and the common ratio is one fifth.

we have to find the sum

$\text{Common ratio}=r=\frac{1}{5}<1$ 

If $r^2<1$ infinite series converges, otherwise it diverges.

Since the sum of any geometric sequence is:

$S_n=\frac{a(1-r^n)}{(1-r)}$

whenever $r^2<1$ the sum of the infinite series is

$S_n=\frac{a}{(1-r)}$

Since a=84 and $r=\frac{1}{5}$ the sum of infinite series

$S_n=\frac{84}{(1-\frac{1}{5})}$

    $=\frac{84}{\frac{4}{5}}$

    $=\frac{5\times84}{4}=105$

Hence, the sum is 105