The population of a local species of bees can be found using an infinite geometric series where a1=860 and the common ratio is 1/5. write the sum in stigma notation and calculate the sum that will be the upper limit of this population

Answer: 1,075

Explanation:

A geometric series means that each term is equal to the previous terms times the common ratio.

This is Aₓ = Aₓ₋₁ r

Given A₁ = 860 and r = (1/5) you have:

A₁ = 860

A₂ = 860 (1/5) ¹

A₃ = 860 (1/5) ²

A₄ = 860 (1/5) ³

A₅ = 860 (1/5) ⁴

...

Aₓ = 860 (1/5) ˣ⁻¹

And the sum of the terms is:

∞

∑ (860) (1/5) ⁿ⁻¹ = 1,075.

n = 1

For that you can use the formula for the sum of an ininite series (when |r| < 1):

∞

∑ Arⁿ = A / (1 - r)

n = 1

Therefore, with A = A₁ = 860, and r = 1/5

∞

∑ (860) (1/5) ⁿ⁻¹ = 860 / [ (1 - (1/5) ] = 860 / (4/5) = 860 * 5 / 4 = 1,075.

n = 1

That is the answer: 1,075.