Problem 44 - Algebra 2 and Trigonometry

Question

Hui started a new job with a weekly salary of \(\$ 400 .\) After one year, and for each year that followed, his salary was increased by 10\(\% .\) Hui left this job after six years. a. List the weekly salary that Hui earned each year. b. Write a recursive definition for this sequence.

Step-by-Step Solution

Answer

a. Yearly salaries: $400, $440, $484, $532.40, $585.64, $644.20. b. Recursive definition: \( S_1 = 400 \); \( S_n = S_{n-1} \times 1.10 \).

Step 1: Initial Salary

Hui started with an initial weekly salary of $400. This is the salary for Year 1.

Step 2: Calculate Salary for Year 2

To find the salary for Year 2, increase the Year 1 salary by 10%. This can be calculated using the formula: \[ S_2 = S_1 + 0.10 \times S_1 = S_1 \times 1.10 \] Substitute \( S_1 = 400 \) to get:\[ S_2 = 400 \times 1.10 = 440 \] So, the weekly salary for Year 2 is $440.

Step 3: Calculate Salary for Year 3

Increase the Year 2 salary by 10% to find the salary for Year 3:\[ S_3 = S_2 \times 1.10 = 440 \times 1.10 = 484 \] The weekly salary for Year 3 is $484.

Step 4: Calculate Remaining Yearly Salaries

Continue increasing each year's salary by 10% for the remaining years:- **Year 4**: \[ S_4 = S_3 \times 1.10 = 484 \times 1.10 = 532.4 \]- **Year 5**: \[ S_5 = S_4 \times 1.10 = 532.4 \times 1.10 = 585.64 \]- **Year 6**: \[ S_6 = S_5 \times 1.10 = 585.64 \times 1.10 = 644.204 \]

Step 5: List Weekly Salaries

The weekly salaries Hui earned each year are: - Year 1: $400 - Year 2: $440 - Year 3: $484 - Year 4: $532.40 - Year 5: $585.64 - Year 6: $644.20

Step 6: Recursive Definition

To express these salaries recursively, define the initial salary and growth rule:- **Base case**: \( S_1 = 400 \)- **Recursive rule**: \( S_{n} = S_{n-1} \times 1.10 \) for \( n > 1 \)

Key Concepts

Initial Value

In any sequence or pattern of numbers, the starting point is crucial. Here, Hui's initial value is his starting weekly salary, which is $400. This initial value provides a foundational point from which all subsequent values of the sequence are calculated.

• An initial value acts like a reference point. For Hui, it meant determining his future salaries based on this base amount.
• The initial value is central in recursive sequences as the calculation of subsequent terms depends on it.

Understanding the initial value makes it easier to comprehend growth over time, as each subsequent calculation begins from this baseline. In any practical financial or mathematical scenario, identifying and defining the initial value is the first step to gaining insights into future trends.

Percent Increase

A percent increase is a way to describe how much a quantity grows relative to an initial amount. For Hui's salary, the percent increase is 10%, meaning every year, his salary grows by 10% from its previous value.

• The percent increase uses a simple formula: Increment value = Initial value \(\times\) (percent increase / 100).
• In Hui's case, the first increase took his salary from \(\$400\) to \(\#440\) (400 \(\times 0.10 + 400\)).
• This method of growth reflects a common financial scenario where earnings, investments, or even costs rise by a steady percentage rate.

Identifying the percent increase helps students understand how incremental growth accumulates over time. It illustrates how continuous percentage increases can lead to exponential growth, a concept frequently encountered in finance and economics.

Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is multiplied by a fixed, non-zero number known as the common ratio. In Hui's case, his salary forms a geometric sequence with a common ratio of 1.10.

• Each year, his salary grows by multiplying it by this ratio, which incorporates the base salary and the percent increase.
• For Hui, starting from \(\$400\), each subsequent salary is calculated as previous salary \(\times 1.10\).
• The recursive definition of the sequence captures the ongoing multiplicative process, succinctly expressed as \( S_n = S_{n-1} \times 1.10 \).

Geometric sequences are crucial for understanding escalating growth patterns. This holds especially in finance, such as in calculating compound interest, where similar sequential multiplication occurs.