Problem 76 - Algebra and Trigonometry

Question

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises \(71-76\). A pendulum swings through an arc of 16 inches. On each successive swing, the length of the arc is \(96 \%\) of the previous length. $$ \begin{array}{cccc} {16,} & {0.96(16),} & {(0.96)^{2}(16),} & {(0.96)^{3}(16)} \\ {\text { Ist }} & {2 \text { nd }} & {3 \text { rd }} & {4 \text { th }} \\ {\text { swing }} & {\text { swing }} & {\text { swing }} & {\text { swing }} \end{array} $$ After 10 swings, what is the total length of the distance the pendulum has swung?

Step-by-Step Solution

Answer

The total length of the distance the pendulum has swung after 10 swings is approximately 155.783 inches.

Step 1: Identify the initial terms

First, identify the first term 'a' of the sequence which is 16, and the common ration 'r', which is 0.96. Thus, a=16 and r=0.96

Step 2: Find The Nth Term

In a geometric sequence, the number of terms \(n\) is given in the problem as 10. That means we are interested in the sum of first 10 terms.

Step 3: Apply the formula

The sum \(S\) of the first \(n\) terms of a geometric sequence can be calculated using the following formula: \[ S = \frac{a \cdot (r^n - 1)}{r - 1} \] Now, plug the values a=16, r=0.96, and n=10 into the formula to find the sum. Let's do the calculation.

Step 4: Calculate the sum

Substituting the values in the formula, we get \[ S = \frac{16 \cdot (0.96^{10} - 1)}{0.96 - 1} \] When calculated, the sum rounds to approximately 155.783 inches.

Key Concepts

Geometric Progression

The concept of a geometric progression, also known as a geometric sequence, is fundamental in understanding how quantities grow or decay at a consistent rate per term. A geometric progression is a sequence 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. For instance, in the exercise related to the pendulum swing, we start with an arc length of 16 inches which is our first term or 'a'. Each subsequent swing is 96% of the length of the previous swing, which establishes our common ratio 'r' as 0.96.

Geometric progressions are ubiquitous in various fields like finance, biology, physics, and computer science, as they can model exponential growth or decay. They are also instrumental in calculating compound interest, population growth, radioactive decay, and much more.

Sum of Geometric Series

The sum of a geometric series is the total value when all the terms of a geometric progression are added together. This sum can be finite or infinite depending on the common ratio and the number of terms. For finite series, we use the formula \[ S = \frac{a(r^n - 1)}{r - 1} \]where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms in the progression.

Applying this formula to our 10-term pendulum swing problem provides us with a structured approach to find the total distance the pendulum swings after 10 iterations. By inserting the known values into the formula, we obtain the total sum which is key in understanding the cumulative effect of the repeating pattern demonstrated by the pendulum's swings.

Convergence of Geometric Series

Convergence in the context of geometric series refers to the behavior of the series as the number of terms goes to infinity. A geometric series converges if the absolute value of the common ratio is less than 1, which would cause the terms to get progressively smaller, approaching zero. Conversely, the series diverges if the common ratio is greater than or equal to 1, leading to terms that grow without bound or oscillate without settling to a point.

In our pendulum example, since the common ratio is 0.96—a number less than 1—the series will converge if we continue to extend the number of swings to infinity. Mathematically, this reveals the fascinating aspect that even as the pendulum continues to swing perpetually, there exists a finite total distance it will cover, which can be described by the sum of an infinite geometric series.