Question

The population of a local species of beetle can be found using an infinite geometric series where a1 = 880 and the common ratio is one fourth. Write the sum in sigma notation, and calculate the sum (if possible) that will be the upper limit of this population.

Answer 1

Data:

infinite geometric series

A1 = 880

r = 1 / 4

The sum of a geometric series  in sigma notation is:

  n              1 - r^n
 ∑ Ai = A ----------- ; where A = A1
i = 1             1-r

 When | r | < 1 the infinite sum exists and is equal to:

  ∞               A
 ∑ Ai =   ---------- ; where A = A1
i = 1          1 - r

So, in this case:

  ∞               880
 ∑ Ai =    -------------- = 4 * 880 / 3 = 3520 /3 = 1173 + 1/3
i = 1         1 - (1/4) 

Answer: 1173 and 1/3