Problem 8 - 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). $$ a_{n}=3^{n-1} $$

Step-by-Step Solution

Answer

First five terms: 1, 3, 9, 27, 81; Common ratio: 3.

Step 1: Identify the General Term

The general formula for the sequence is given as \(a_{n} = 3^{n-1}\). This formula will be used to determine the specific terms for any given \(n\).

Step 2: Calculate the First Term

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

Step 3: Calculate the Second Term

Substitute \(n = 2\) into the formula: \(a_{2} = 3^{2-1} = 3^{1} = 3\). Thus, the second term is \(3\).

Step 4: Calculate the Third Term

Substitute \(n = 3\) into the formula: \(a_{3} = 3^{3-1} = 3^{2} = 9\). Thus, the third term is \(9\).

Step 5: Calculate the Fourth Term

Substitute \(n = 4\) into the formula: \(a_{4} = 3^{4-1} = 3^{3} = 27\). Thus, the fourth term is \(27\).

Step 6: Calculate the Fifth Term

Substitute \(n = 5\) into the formula: \(a_{5} = 3^{5-1} = 3^{4} = 81\). Thus, the fifth term is \(81\).

Step 7: Identify the Common Ratio

The common ratio \(r\) in a geometric sequence is the factor multiplied to get from one term to the next. Calculate \(r\) using the first two terms: \(r = \frac{a_{2}}{a_{1}} = \frac{3}{1} = 3\). So, the common ratio is \(3\).

Step 8: Graph the Sequence Terms

Graph the terms (1, 3, 9, 27, 81) on a coordinate plane with the horizontal axis representing the term number and the vertical axis representing the value of the term. Each term corresponds to a point on the graph: (1,1), (2,3), (3,9), (4,27), (5,81).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key element that determines how the sequence progresses. This ratio, denoted as \( r \), is the constant factor you multiply by to move from one term in the sequence to the next.

In our sequence, starting with the formula \( a_n = 3^{n-1} \), the common ratio can be found by dividing any term by the previous term. For example:

• From the first term to the second term: \( r = \frac{3}{1} = 3 \)
• From the second term to the third term: \( r = \frac{9}{3} = 3 \)
• This consistent value confirms that the common ratio \( r \) is indeed 3.

Once the common ratio is known, it offers a swift method to determine the next terms in the sequence without recalculating from the general formula.

• If you know the previous term: simply multiply it by the common ratio.
• This makes predicting the sequence or checking existing values much simpler.

Understanding the common ratio can also deepen comprehension of how the sequence behaves, including how rapidly it grows.

Graphing Sequences

Graphing a geometric sequence provides a visual representation of how the terms progress. To graph a sequence like \( a_n = 3^{n-1} \), plot each term against the term number on a coordinate plane.

In this exercise:

• The horizontal axis represents the term number \( n \).
• The vertical axis represents the value of the term \( a_n \).

For our sequence, the data points are:

• (1, 1)
• (2, 3)
• (3, 9)
• (4, 27)
• (5, 81)

When plotted, these points will appear to rise steeply because the sequence grows exponentially as \( n \) increases. Connecting the points with a smooth curve can help visualize the rate of growth and the nature of the sequence.

Graphing helps in:

• Understanding the overall trend of the sequence.
• Identifying how rapidly the values increase.

This method solidifies abstract numerical sequences into a concrete form that's easier to explore.

Geometric Progression

A geometric progression or sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This type of sequence is characterized by exponential growth or decay.

In our case, the sequence given by \( a_n = 3^{n-1} \) is a perfect example of geometric progression:

• Each term results from multiplying the previous term by the common ratio \( 3 \).
• The sequence starts at 1 and grows to 3, 9, 27, 81, and so on.

The nature of geometric progression means:

• The terms increase or decrease exponentially rather than linearly.
• This can result in very large or very small numbers quickly, depending on \( r \).

Analyzing such sequences is crucial for understanding concepts in both mathematics and real-world applications, such as calculating compound interest or population growth.

Geometric sequences also highlight the power of exponential functions and emphasize the importance of knowing the common ratio, which dictates how the sequence behaves.