The Environmental Protection Agency (EPA) determines that...

The Environmental Protection Agency (EPA) determines that Maple Lake has 250 tons of pollutant as a result of industrial waste and that $10 \%$ of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution control laws that result in 15 tons of new pollutant entering the lake each year. The amount of pollutant in the lake after $n$ years is given by the recursively defined sequence $$ p_0=250 \quad p_n=0.9 p_{n-1}+15 $$ Determine the amount of pollutant in the lake after 2 years. That is, determine $p_2$.

The Environmental Protection Agency (EPA) determines that Maple Lake has 250 tons of pollutant as a result of industrial waste and that $10 \%$ of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution control laws that result in 15 tons of new pollutant entering the lake each year. The amount of pollutant in the lake after $n$ years is given by the recursively defined sequence
$$
p_0=250 \quad p_n=0.9 p_{n-1}+15
$$
Determine the amount of pollutant in the lake after 2 years. That is, determine $p_2$.

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Key Concepts

Recurrence Relation

A recurrence relation defines each term of a sequence as a function of one or more prior terms. It is essential in situations where a current state depends on the previous state, allowing for the step-by-step buildup of the sequence.

Initial Condition

The initial condition specifies the starting point of the sequence. It is the cornerstone for all subsequent calculations in a recurrence relation, as every future term is determined based on this initial value and the recurrence rule.

Linear Difference Equation

A linear difference equation is a type of recurrence relation where each term is a linear combination of previous terms plus a constant. This structure allows for modeling systems with both decay and constant external contributions.

Exponential Decay Factor

The exponential decay factor in a recurrence relation, represented by a multiplier less than one, illustrates the proportional reduction of a quantity over time. This concept is widely used to model processes where some percentage of a substance diminishes during each time interval.

Iterative Computation

Iterative computation involves calculating each term in a sequence by repeatedly applying the recurrence relation. It is a step-by-step process used to determine the state of a system after a finite number of intervals.

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