Problem 76 - College Algebra

Question

A pendulum swings through an are of 16 inches. On each successive swing, the length of the arc is \(96 \%\) of the previous length. After 10 swings, what is the total length of the distance the pendulum has swung?

Step-by-Step Solution

Answer

Use your calculated value for \(S_{10}\) from Step 3 as the short answer.

Step 1: Identify the first term, common ratio, and total terms

The first term \(a\) of our geometric series is 16 inches, the common ratio \(r\) is \(96 \%\) or 0.96, and the total terms \(n\) is 10. So, \(a = 16\), \(r = 0.96\), and \(n = 10\).

Step 2: Substitute the values into the geometric series formula

Using the formula \(S_n = \frac{a(1-r^n)}{1-r}\), substitute the values for \(a\), \(r\), and \(n\): \(S_{10} = \frac{16(1-0.96^{10})}{1-0.96}\).

Step 3: Solve the equation

Calculate the expression inside the parentheses first (following the order of operations/bodmas rule), then solve for \(S_{10}\).

Step 4: Interpret the result

The value obtained for \(S_{10}\) will be the total distance the pendulum has swung after 10 swings.

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term, after the first, is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This special type of sequence is often found in nature and in mathematical applications. To understand this better, imagine a pendulum swinging through a series of arcs. Each swing represents a term in the sequence. The length of each successive swing forms a geometric sequence because the length of the swing is multiplied by the same number, known as the common ratio, compared to the previous swing.

Finding the sum of a geometric sequence involves understanding each component:

• First term (\(a\)): The initial length of the swing or starting point.
• Common ratio (\(r\)): Multiplier of each term in the sequence.
• Total terms (\(n\)): The number of swings or terms to consider.

In many problems, you might need to find the sum of such sequences, like the total length of the pendulum's swings over several terms. By knowing the first term, the common ratio, and the total terms, you can determine the sum.

Common Ratio

The common ratio is a critical concept in understanding geometric sequences. It is the factor that each term is multiplied by to get the next term in the sequence. Mathematically, if you have a geometric sequence with terms \(a_1, a_2, a_3, \ldots\), then the common ratio \(r\) is defined as \(a_2 / a_1 = a_3 / a_2 = \ldots\).

In the context of the pendulum's swing, the common ratio is 0.96, meaning that each successive swing covers 96% of the length of the previous swing. This idea stems from reducing or scaling down the length of the swing in a consistent manner each time.

Understanding the common ratio can help you:

• Predict the next terms in the sequence easily.
• Calculate the total sum of the sequence efficiently.
• Apply the concept to physical phenomena, like decay rates or diminishing lengths.*

Knowing the common ratio allows you to model real-world scenarios with geometric sequences effectively.

Arithmetic Progression

While arithmetic progression might seem similar to geometric sequences, there is a fundamental difference. An arithmetic progression, also known as an arithmetic sequence, is a number pattern where each term after the first is obtained by adding a constant value, known as the common difference, to the previous term. In contrast, geometric sequences multiply by a constant ratio.

For example, if you have an arithmetic sequence starting at 2 and increasing by 3—like 2, 5, 8, 11—each term is obtained by adding 3 to the previous term. With arithmetic sequences:

• Each term is obtained by addition.
• The sequence has a linear pattern.

In practical scenarios, arithmetic progressions can model situations where growth or change happens uniformly, like saving constant sums of money each month. Although the pendulum swing problem deals with geometric sequences, it's helpful to contrast with arithmetic progressions for a complete understanding of sequence types.