Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

Graphing the Terms of a Sequence Use a graphing utility to graph the first 10 terms of the sequence. $$a_{n}=12(-0.75)^{n-1}$$

Step-by-Step Solution

Answer

The first 10 terms of the sequence and their graph reveal a geometric sequence shrinking in absolute value, alternating between positive and negative.

Step 1: Understand the Sequence

The sequence given is a geometric sequence, as each term is the product of the previous term and a constant. The constant in this case is -0.75 and the first term, \(a_1\), is 12.

Step 2: Calculate the First 10 Terms

Using the formula \(a_n = a_1 \times r^{(n-1)}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number, calculate the first 10 terms of the sequence. For example, the second term, \(a_2\), would be calculated as \(12 \times (-0.75)^{2-1} = 12 \times -0.75 = -9\). Repeat this for \(n\) from 1 to 10.

Step 3: Plotting the Terms

Use a graphing utility to plot these calculated points. The term number \(n\) as the x-value and the term value \(a_n\) as the y-value, then connect the points.

Key Concepts

Graphing Sequences

Graphing sequences is a fantastic way to visualize mathematical patterns. In a geometric sequence, graphing involves plotting each sequence term on a coordinate system. Here, the term number acts as the x-value while the term value itself becomes the y-value. For instance, in the sequence \( a_n = 12(-0.75)^{n-1} \), to plot the first 10 terms, each \( n \) is your x-axis value, and each corresponding \( a_n \) is your y-axis value.When you graph these points, you can see how the sequence behaves. Geometric sequences typically show exponential growth or decay. As you plot consecutive terms:

• The points create a curve or steep line.
• This visual representation highlights features like the direction of the sequence (increasing or decreasing).

Use technology, like a graphing calculator or software, to quickly and accurately graph sequences. It enhances understanding by clearly displaying patterns and trends.

Common Ratio

The common ratio is a hallmark of geometric sequences. It determines how each term in the sequence relates to its predecessor. In the sequence \( a_n = 12(-0.75)^{n-1} \), the common ratio is \(-0.75\). This value is crucial because:

• It indicates multiplication applied to each term to get the next term.
• Determines whether the sequence shows growth (if ratio > 1) or decay (if ratio \(< 1\)).

Since the ratio here is negative, it also affects direction. This flips the sign of each term relative to the last. Understanding the effect of the common ratio helps in predicting the sequence's behavior without calculating each term. This insight is useful for complex sequences beyond initial terms.

Sequence Terms

The terms of a sequence are like a building block, each representing a position in an ordered list. Understanding the calculation and role of each term in a geometric sequence enhances comprehension. Let's revisit the sequence \( a_n = 12(-0.75)^{n-1} \).To find specific sequence terms, use the general formula \( a_n = a_1 \times r^{(n-1)} \). This tells us:

• \( a_1 \), the first term, sets the starting point of the sequence. Here, it's 12.
• Each successive term is derived by multiplying the previous term by \(-0.75\).
• The position, \( n \), dictates the exponent applied to the common ratio.

Calculating terms one by one gives a step-by-step guide to predict how the sequence unfolds. This orderly process ensures each sequence term contributes to the overall pattern and behavior, providing clarity and predictability.