Problem 2 - Algebra and Trigonometry

Question

The \(n\)th term of a sequence is given. (a) Find the first five terms of the sequence. (b) What is the common ratio \(r ?\) (c) Graph the terms you found in (a). $$ a_{n}=3(-4)^{n-1} $$

Step-by-Step Solution

Answer

The first five terms are 3, -12, 48, -192, and 768. The common ratio is -4. Graph the terms on a coordinate plane with term numbers on the x-axis.

Step 1: Finding the First Term

To find the first term of the sequence, substitute \(n = 1\) into the formula \(a_n = 3(-4)^{n-1}\): \[a_1 = 3(-4)^{1-1} = 3(-4)^0 = 3(1) = 3.\] Thus, the first term is 3.

Step 2: Finding the Second Term

Substitute \(n = 2\) into the formula:\[a_2 = 3(-4)^{2-1} = 3(-4)^1 = 3(-4) = -12.\]Thus, the second term is -12.

Step 3: Finding the Third Term

Substitute \(n = 3\) into the formula:\[a_3 = 3(-4)^{3-1} = 3(-4)^2 = 3(16) = 48.\]Thus, the third term is 48.

Step 4: Finding the Fourth Term

Substitute \(n = 4\) into the formula:\[a_4 = 3(-4)^{4-1} = 3(-4)^3 = 3(-64) = -192.\] Thus, the fourth term is -192.

Step 5: Finding the Fifth Term

Substitute \(n = 5\) into the formula:\[a_5 = 3(-4)^{5-1} = 3(-4)^4 = 3(256) = 768.\]Thus, the fifth term is 768.

Step 6: Determine the Common Ratio

The common ratio \(r\) in a geometric sequence can be found by dividing the second term by the first term.\[r = \frac{a_2}{a_1} = \frac{-12}{3} = -4.\] Therefore, the common ratio is \(-4\).

Step 7: Graphing the Terms

To graph the sequence terms found in (a), plot the points \((1, 3), (2, -12), (3, 48), (4, -192), (5, 768)\) on a coordinate plane with the term number on the x-axis and the term value on the y-axis. Connect the points smoothly to show the rapid increase in their absolute values.

Key Concepts

Understanding the Common Ratio in Geometric Sequences

In a geometric sequence, the common ratio is a critical concept that defines how the sequence progresses from one term to the next. It is a constant factor that each term is multiplied by to get the subsequent term in the sequence.
For the sequence described by formula \(a_n = 3(-4)^{n-1}\), we can determine the common ratio \(r\) by dividing any term by the previous term. This ratio between consecutive terms remains constant throughout the sequence.
In our example, the common ratio \(r\) is found as \(r = \frac{a_2}{a_1} = \frac{-12}{3} = -4\). This reveals that each term is obtained by multiplying the previous term by \(-4\).
Understanding the common ratio helps us grasp the rapid changes in the sequence, indicating the use of multiplication by a consistent factor. This is particularly helpful when predicting future terms without manual computation.

Graphing Sequences for Better Visualization

Graphing a sequence provides a visual representation of how the terms progress. In our example sequence, an insightful way to understand the nature of the infinite terms is to see them on a graph.
The sequence \(a_n = 3(-4)^{n-1}\) can be graphed by plotting points corresponding to the terms we calculated. For instance, the first five terms are plotted as

• (1, 3)
• (2, -12)
• (3, 48)
• (4, -192)
• (5, 768)

on a coordinate plane.
Place each term number on the x-axis and its respective value on the y-axis. Connecting these points smoothly will illustrate the exaggerated swings caused by the common ratio \(-4\), resulting in alternating large positive and negative values. This visual aid can greatly improve understanding by showing the shocking yet predictable nature of geometric sequences.

Identifying Terms of a Sequence

To identify particular terms in a geometric sequence, you must understand the sequence's formula, which in our case is \(a_n = 3(-4)^{n-1}\). The "term of a sequence" refers to the position or number in the sequence that you are trying to find. Each term indicates its spot by index \(n\), showing how each value proceeds according to the sequence formula.
By substituting different values of \(n\) into the formula, we can conveniently determine specific terms:

• For \(n = 1\), \(a_1 = 3\)
• For \(n = 2\), \(a_2 = -12\)
• For \(n = 3\), \(a_3 = 48\)
• For \(n = 4\), \(a_4 = -192\)
• For \(n = 5\), \(a_5 = 768\)

Understanding how to find each term helps elucidate the overall pattern and relationship governed by the sequence's common ratio, making it easier to extend the sequence to any desired number of terms.