Problem 16 - Beginning and Intermediate Algebra

Question

Find the general term, \(a_{m}\) for each geometric sequence. Then, find the indicated term. $$a_{1}=3, r=8 ; a_{3}$$

Step-by-Step Solution

Answer

The general term of the given geometric sequence is \(a_m = 3 \times 8^{m-1}\). The third term, \(a_3\), is found by substituting \(m = 3\) into the formula, resulting in \(a_3 = 3 \times 8^2 = 3 \times 64 = 192\).

Step 1: Understanding the geometric sequence formula

A geometric sequence 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. The general term of a geometric sequence can be written as: $$a_m = a_1 \times r^{m-1}$$ Where \(a_m\) is the \(m\)th term, \(a_1\) is the first term, \(r\) is the common ratio, and \(m\) is the term number.

Step 2: Find the general term of the given geometric sequence

In this exercise, we are given \(a_1 = 3\), \(r = 8\). We can find the general term \(a_m\) by substituting these values into the formula: $$a_m = 3 \times 8^{m-1}$$

Step 3: Find the specific term \(a_3\)

Now that we have the general term, we can find \(a_3\) by substituting \(m = 3\) into the formula: $$a_3 = 3 \times 8^{3-1}$$

Step 4: Evaluate the expression

Now, we'll simplify the expression to get the value of \(a_3\): $$a_3 = 3 \times 8^{2}$$ $$a_3 = 3 \times 64$$ $$a_3 = 192$$ So, the general term of the geometric sequence is \(a_m = 3 \times 8^{m-1}\) and the value of the third term, \(a_3\), is \(192\).

Key Concepts

Understanding the Geometric Sequence Formula

A geometric sequence is a series of numbers where each term is obtained by multiplying the previous term by a constant value, known as the common ratio. This relationship is expressed with the geometric sequence formula: \[ a_m = a_1 \times r^{m-1} \]In this formula:

• \(a_m\) represents the \(m\)th term we wish to find.
• \(a_1\) is the first term in the sequence.
• \(r\) is the common ratio, which remains consistent throughout the series.
• \(m\) is the specific term number.

This equation allows you to determine any term within the geometric sequence by substituting for \(a_1\), the common ratio \(r\), and the position \(m\).
Once you have these values, simply plug them into the formula and perform the necessary calculations.

Exploring the Common Ratio

The common ratio \(r\) is a crucial component of geometric sequences. It is the constant factor that each term is multiplied by to yield the next term in the sequence.
To find the common ratio, you can divide any term by the preceding term:\[ r = \frac{a_{m+1}}{a_m} \]The common ratio can be a positive or negative number and could be greater than, equal to, or less than one:

• If \(r > 1\), the sequence is increasing.
• If \( 0 < r < 1\), the sequence is decreasing.
• If \(r = 1\), the sequence stays constant.
• If \(r\) is negative, terms alternate in sign.

In the given exercise, the common ratio is 8, indicating that each term multiplies by 8 to arrive at the next value.

Determining the General Term of a Geometric Sequence

The general term, sometimes called the "nth term," describes the position of any term in the sequence. Knowing this term is essential because it allows you to find any term without having to manually calculate all the previous ones.
By applying the geometric sequence formula, \[ a_m = a_1 \times r^{m-1} \], we simplify calculating any desired term for given parameters \(a_1\), \(r\), and \(m\).
In the exercise, you identified \(a_1 = 3\), and \(r = 8\). These values help to derive the general formula for this sequence:\[ a_m = 3 \times 8^{m-1} \]This formula signifies you can now calculate any specific term by substituting \(m\) with the term position.

Sequence Evaluation and Finding Specific Terms

After establishing the general term, you can quickly compute any specific term by plugging the corresponding term number into the general formula. For example, to find the third term \(a_3\), you substitute \(m = 3\) into \[ a_m = 3 \times 8^{m-1} \]This becomes: \[ a_3 = 3 \times 8^{3-1} \] Calculating further:

• Calculate the power: \(8^{2} = 64\).
• Multiply the result by the first term \(3\): \(a_3 = 3 \times 64 = 192\).

Thus, the third term in this sequence is 192. Sequence evaluation like this not only finds a specific term quickly but also validates the correct implementation of the geometric sequence formula.