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When we talk about exponential growth, we are referring to a pattern of data that shows greater increases over time, creating a curve on a graph that starts out slowly and then rises steeply. This kind of growth occurs in various domains, including biology, finance, and technology. In the context of our example concerning a baseball player's salary, exponential growth is observed because the salary increases by a fixed percentage each year, leading to larger and larger increases as time goes on.

In mathematics, exponential growth is described by the formula
\[ P(t) = P_0 \times (1+r)^t \]
where:

• \(P_0\) is the initial amount,
• \(r\) is the growth rate (expressed as a decimal), and
• \(t\) is the time period.

This formula is pivotal in understanding how investments, populations, and salaries like that of the athlete's can grow over the years. It becomes apparent that even a seemingly modest annual increase can lead to a significantly higher value over an extended time due to the compound effect of exponential growth.

A mathematical sequence is a list of numbers arranged in a specific order following a particular rule. Sequences can be finite or infinite and are commonly categorized as arithmetic, geometric, or other types based on the pattern they follow. In the case discussed, we deal with a geometric sequence, where each term after the first is found by multiplying the previous term by a constant called the common ratio.

For a geometric sequence, the nth term can be calculated using the formula:
\[ a_n = a_1 \times r^{(n-1)} \]
where:

• \(a_1\) is the first term of the sequence,
• \(r\) is the common ratio, and
• \(n\) is the term's position in the sequence (with \(n=1\) being the first term).

The sequence is a powerful concept as it helps us model situations where quantities grow by a constant factor, which is precisely what happens with the athlete's contract where his salary grows by a fixed rate of \(4\text{%}\) annually.

The term percentage increase refers to the rate at which a quantity grows relative to its previous value, expressed in percentage. It's a common way to describe changes in everything from prices to salaries and is calculated by the formula:
\[ \text{Percentage Increase} = \frac{\text{Increase}}{\text{Original Number}} \times 100 \text{%} \]
In the context of the initial problem, the percentage increase is the \(4\text{%}\) annual raise in the athlete's salary. This kind of increase can be deceiving because, as it compounds over time, the actual monetary increase from one year to the next becomes larger, even though the percentage remains the same.

Understanding how percentage increases work in geometric sequences, like the athlete’s salary, helps in planning financial futures, analyzing economic trends, and making informed decisions about investments and contracts. The consistent \(4\text{%}\) increase is small on paper, but over time, it has a profound compound effect, making the athlete’s seventh year of income substantially greater than the first year.