Find the number of units of the molecule in the river after n weeks

[Imuell1]

Homework Statement

A chemical plant produces pesticide that contains a molecule potentially harmful to people if the concentration is too high. The plant flushes out the tanks containing the pesticide once a week, and the discharge flows into the river. The molecule breaks down gradually in water so that 90% of the amount remaining each week is dissipated by the end of the next week. Suppose that D units of the molecule are discharged each week.

a) Find the number of units of the molecule in the river after n weeks.
b) Estimate the amount of the molecule in the water supply over a very long time(hint find the sum of the series)
c) If the toxic level of the molecule is T units, how large an amount of the molecule can the plant discharge each week?

The Attempt at a Solution

A)
I got a₁=D a₂=D+.01(D) a₃=D+.01(D+.01D)

so a_n=D+.01(a_n-1) or a_n=a₁+.01(a_n-1)

B) I'm not sure if I did part b right and I got the sum as n=0 to \infty of a₁+.01(a_n-1) but I have a strong feeling this isn't right.

C) I don't know what to do for part C

 

[HallsofIvy]

Homework Statement

A chemical plant produces pesticide that contains a molecule potentially harmful to people if the concentration is too high. The plant flushes out the tanks containing the pesticide once a week, and the discharge flows into the river. The molecule breaks down gradually in water so that 90% of the amount remaining each week is dissipated by the end of the next week. Suppose that D units of the molecule are discharged each week.

a) Find the number of units of the molecule in the river after n weeks.
b) Estimate the amount of the molecule in the water supply over a very long time(hint find the sum of the series)
c) If the toxic level of the molecule is T units, how large an amount of the molecule can the plant discharge each week?

The Attempt at a Solution

A)
I got a₁=D a₂=D+.01(D) a₃=D+.01(D+.01D)

so a_n=D+.01(a_n-1) or a_n=a₁+.01(a_n-1)

This is a "recursive" equation but it does not answer the question- you have not yet found a formula for a_n. You say that a₃= D(1+ .01+ .01²). Do you see that that is a geometric series? What is the formula for the sum of a finite geometric series?

B) I'm not sure if I did part b right and I got the sum as n=0 to \infty of a₁+.01(a_n-1) but I have a strong feeling this isn't right.

Well, what is that sum? What is the formula for the sum of an infinite geometric series?

C) I don't know what to do for part C

For what values of D is the answer to (B) less than T?