Question

The population of a local species of dragonfly can be found using an infinite geometric series where a1 = 30 and the common ratio is 2/5. Write the sum in sigma notation and calculate the sum that will be the upper limit of this population.

Answer 1

Answer:

50 is the answer.

Step-by-step explanation:

We have, $a_{1} =30$ and common ratio d=2/5.

So, the general form of the geometric series is $a_{n} =a_{1} *d^{n-1}$, for 'n' is from 1 to infinity.

Hence, the sum in sigma form = $\sum a_{1} *d^{n-1}$, where n goes from 1 to infinity.

Now, the infinite sum of geometric series = $\frac{a_{1} }{1-d}$

i.e. $\frac{30}{2/5}$ = 50

Hence, the sum which will be the upper limit is 50

Answer 2

The sum will be: 

sigma(i = 1 to infinity, 30*(2/5)^i) 

->$\lim_{i \to \infty} \frac{30(1- \frac{2}{5}^{i} )}{ \frac{3}{5} }$

Which is equal to 30/(3/5) = 50