You have just signed a 7-year professional football league...

You have just signed a 7-year professional football league contract with a beginning salary of $$\$ 2,000,000$$ per year. Management gives you the following options with regard to your salary over the 7 years. 1. A bonus of $$\$ 100,000$$ each year 2. An annual increase of $4.5 \%$ per year beginning after 1 year 3. An annual increase of $$\$ 95,000$$ per year beginning after 1 year Which option provides the most money over the 7-year period? Which the least? Which would you choose? Why?

You have just signed a 7-year professional football league contract with a beginning salary of $$\$ 2,000,000$$ per year. Management gives you the following options with regard to your salary over the 7 years.
1. A bonus of $$\$ 100,000$$ each year
2. An annual increase of $4.5 \%$ per year beginning after 1 year
3. An annual increase of $$\$ 95,000$$ per year beginning after 1 year
Which option provides the most money over the 7-year period? Which the least? Which would you choose? Why?

Key Concepts

Comparing Compensation Options

Evaluating various compensation options involves analyzing how different growth patterns—such as fixed bonuses, percentage-based increases, or constant increments—affect total earnings. This comparison is central to decision-making in financial planning and contract negotiation, enabling one to choose the most beneficial option based on long-term outcomes.

Geometric Sequence

A geometric sequence is a series of numbers where each term is obtained by multiplying the previous term by a constant factor. For salary increases based on a percentage (compound growth), the value grows exponentially over time, which can lead to significantly higher totals compared to a fixed increment over several periods.

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which each term increases by a constant amount. In the context of salary increases or fixed bonuses, the increase from year to year is the same, making it straightforward to calculate the total sum over a period using the arithmetic series formula.

Summation of Series

Summing a series of payments over multiple periods requires identifying whether the growth pattern is arithmetic or geometric. The use of summation formulas for these sequences is a fundamental concept in financial calculations as it allows for the precise comparison of different payment structures over time.

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