Problem 20 - Beginning and Intermediate Algebra

Question

Find the general term, \(a_{m}\) for each geometric sequence. Then, find the indicated term. $$a_{1}=7, r=3 ; a_{5}$$

Step-by-Step Solution

Answer

The general term, \(a_{m}\), for the geometric sequence is \(a_{m} = 7 \cdot 3^{m-1}\), and the indicated term, \(a_{5}\), is 567.

Step 1: Identify given values

In the given problem, we know the first term \(a_{1} = 7\) and the common ratio \(r = 3\).

Step 2: Calculate the general term

To find the general term \(a_{m}\), we use the formula for the general term of a geometric sequence: $$a_{m} = a_{1} \cdot r^{m-1}$$ Substitute the values of \(a_{1}\) and \(r\) into the formula: $$a_{m} = 7 \cdot 3^{m-1}$$

Step 3: Find the indicated term

We want to find the 5th term of the sequence, so we have to substitute \(m = 5\) into the general term equation: $$a_{5} = 7 \cdot 3^{5-1}$$ Simplify the exponent: $$a_{5} = 7 \cdot 3^{4}$$ Now calculate \(3^4\): $$a_{5} = 7 \cdot 81$$ Finally, multiply 7 by 81: $$a_{5} = 567$$ The general term, \(a_{m}\), for the geometric sequence is: $$a_{m} = 7 \cdot 3^{m-1}$$ The indicated term, \(a_{5}\), is 567.

Key Concepts

General Term

The general term of a sequence is essentially a formula that allows us to find any term in the sequence without listing all previous terms. In the context of a geometric sequence, the general term is used to define the specific formula based on the initial term and the common ratio.
For a geometric sequence, the general term is given by the equation:

• \( a_m = a_1 \cdot r^{m-1} \)

Where:

• \( a_m \) is the term in the sequence you're trying to find,
• \( a_1 \) is the first term of the sequence,
• \( r \) is the common ratio,
• \( m \) is the term number.

By using this formula, you can calculate any term in the sequence, as long as you know the first term and the common ratio. This makes finding any term in a geometric sequence quick and efficient.

Common Ratio

The common ratio in a geometric sequence is a crucial factor that determines how the sequence progresses. It is the factor by which we multiply one term to get the next term.
A geometric sequence showcases consistency in its multiplication pattern, unlike arithmetic progressions where the difference is additive.
For instance, if you have a geometric sequence beginning with 7 and a common ratio of 3, the sequence unfolds as follows:

• The second term is \( 7 \times 3 = 21 \)
• The third term is \( 21 \times 3 = 63 \)
• The fourth term is \( 63 \times 3 = 189 \)

In our exercise, the given common ratio is 3, meaning each term is three times the previous one. This consistency is what makes geometric sequences predictable and allows us to compute any term using the general term formula.

Arithmetic Progression

While the original exercise is about geometric sequences, it's helpful to differentiate them from arithmetic progressions. An arithmetic progression (AP) involves a consistent addition to each term, rather than multiplication. This is crucial in distinguishing between the two.
In an arithmetic progression, the difference between consecutive terms is called the common difference. For example, in the sequence 2, 5, 8, 11, etc., each term increases by 3, representing the common difference.
This concept contrasts with a geometric sequence, where terms change by being multiplied by a fixed number, as illustrated by our exercise's common ratio. Understanding the difference helps in recognizing patterns in sequences and applying the correct formulas.

Exponential Growth

Geometric sequences are closely related to the concept of exponential growth. In fact, exponential growth can be understood as a geometric sequence applied over time where a quantity multiplies by a consistent ratio.
For example, when a population doubles every year, this is exponential growth. Similarly, in our geometric sequence, each term becomes significantly larger than the previous due to multiplication, reflecting exponential characteristics.
The general term formula \( a_m = a_1 \cdot r^{m-1} \) clearly shows the exponential nature. Each term involves raising the common ratio \( r \) to a power, making each subsequent term grow at an increasing rate.
This concept is seen not just in mathematics, but in various real-world scenarios such as finance, biology, and physics, where quantities grow rapidly over time.