The population of a local species of flies can be found using an infinite geometric series where a1 = 940 and the common ratio is one fifth. write the sum in sigma notation, and calculate the sum (if possible) that will be the upper limit of this population.

A. Sigma notation

The formula for finding the nth value of the geometric series is given as:

an = a1 * r^n

Where,

an = nth value of the series

a1 = 1st value in the geometric series = 940

r = common ratio = 1/5

n = nth order

The sigma notation for the sum of this infinite geometric series is therefore,

                     (see attached photo)

B. Sum of the infinite geometric series

The formula for calculating the sum of an infinite geometric series is given as:

S = a1  / (1 – r)

Substituting the given values:

S = 940 / (1 – 1/5)

S = 1,175

If you make the two numbers X and Y then we can make a system of equations:

Y - X = 9
X*Y = 162

Using the first equation you can set Y equal to (9 + X) and then plug that in to the second equation

X*(9 + X) = 162

Then if we multiply that out and subtract 162 from both sides we get a quadratic equation

X^2 + 9X - 162 = 0

From here you can just factor to get:

(X + 18)(X - 9) = 0
X = -18, 9

Then when you solve for Y you’re two numbers are either (-18, -9) or (9, 18).