Problem 60 - College Algebra

Question

Use the formula for the general term (the nth term) of a geometric sequence to solve. Isuppose you Save 1 dollar the first day of a month, 2 dollar the second day, 4 dollar the third day, and so on. That is, each day you save twice as much as you did the day before. You are offered a job that pays 30,000 dollar for the first year with an annual increase of \(5 \%\) per year beginning in the second year. That is, beginning in year \(2,\) your salary will be 1.05 times what it was in the previous year. What can you expect to earn in your sixth year on the job?

Step-by-Step Solution

Answer

The expected earning in the sixth year on the job, by applying the formula for the nth term of a geometric sequence, is calculated from the expression: \(T_6 = 30000 * 1.05^{(6-1)}\)

Step 1: Understanding the problem

In this scenario, a job pays $30,000 per annum with a 5% annual increase starting from the second year. This exercise is to calculate the projected salary for the sixth year. The growth rate is constant (5%). Hence, this scenario can be treated as a geometric sequence problem. Instead of the ratio being multiplier of 2, it is 1.05 (which is equivalent to 5% increase).

Step 2: Formulating the geometric sequence

Given the geometric sequence starts with $30,000 and a common ratio of 1.05 (due to 5% increase), we can denote the sequence as follows: \(a, ar, ar^2, ar^3, ar^4, ar^5, ar^6, ...\), where:- \(a\) is the first term of the sequence, which is $30,000,- \(r\) is the common ratio, which is 1.05.

Step 3: Applying the nth term formula of a geometric sequence

The nth term of a geometric sequence is given by the formula, \(T_n = a * r^{(n-1)}\).Now we can solve for the sixth term (the sixth year's salary) by replacing \(a\) with $30,000, \(r\) with 1.05, and \(n\) with 6.So, \(T_6 = 30000 * 1.05^{(6-1)}\)

Step 4: Calculating the sixth year's salary

Calculate \(T_6 = 30000 * 1.05^{(6-1)}\) to find the salary of the sixth year.

Key Concepts

Geometric Sequence Formula

A geometric sequence is a set of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. The formula for the nth term of a geometric sequence \(T_n\) is expressed as \( T_n = a * r^{(n-1)} \), where \(a\) is the first term and \(r\) is the common ratio. Understanding this formula allows you to calculate any term in the sequence without listing all the preceding terms.

For example, if you save money in a way that your savings double each day (\(r = 2\)), your savings form a geometric sequence. By applying the geometric sequence formula, you can predict how much you'll save on any given day. The ability to predict future values is what makes the geometric sequence an essential concept in various fields including finance and computer science.

Geometric Series

While a geometric sequence refers to the ordered list of numbers that follow a particular pattern, a geometric series is the sum of a geometric sequence. It is another important concept especially when you're dealing with cumulative totals, such as interest over time in banking or the total amount of a drug in the bloodstream after multiple doses.

The sum of the first \(n\) terms of a geometric sequence can be found using the formula \({S_n = a * \frac{1 - r^n}{1 - r}}\) if \(r\) does not equal 1. Here, \(S_n\) is the sum of the first \(n\) terms. This formula is useful when determining totals without having to manually add each term. For instance, rather than adding each day's savings, you could use this formula to calculate the total amount saved over a period.

Exponential Growth

Exponential growth occurs when the increase of a quantity is proportional to its current value. This pattern results in the value of a quantity growing at an ever-increasing rate. It is exemplified in our exercise by the salary increase of 5% each year. The principle of exponential growth underlies the model of a geometric sequence when the common ratio is greater than 1. In simpler terms, it means that something grows by a fixed percentage rather than a fixed amount.

In financial contexts, exponential growth is pivotal in understanding compound interest, investments, and even inflation. It's crucial to comprehend exponential growth because it helps in making long-term decisions. For instance, when analyzing the job offer that includes a 5% increase in salary each year, you can calculate your potential earnings over time, allowing for more informed career and financial planning.