Problem 15 - Beginning and Intermediate Algebra

Question

Find the general term, \(a_{m}\) for each geometric sequence. Then, find the indicated term. $$a_{1}=4, r=7 ; a_{3}$$

Step-by-Step Solution

Answer

The general term of the geometric sequence is \(a_m = 4 \cdot 7^{m-1}\) and the third term is 196.

Step 1: Write the formula for the nth term of a geometric sequence.

The formula for the nth term of a geometric sequence is: \(a_n = a_1 \cdot r^{n-1}\) where \(a_n\) denotes the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

Step 2: Substitute the given values into the formula.

Given the first term (\(a_1 = 4\)), the common ratio (\(r = 7\)), and a general term (\(a_m\)), we can write the formula with these values: \(a_m = 4 \cdot 7^{m-1}\)

Step 3: Find the third term of the sequence using the formula.

To find the third term of the geometric sequence, we will substitute \(m = 3\) into the formula: \(a_3 = 4 \cdot 7^{3-1}\)

Step 4: Calculate the third term of the sequence.

Calculate the value of the exponent in the formula and solve for \(a_3\): \(a_3 = 4 \cdot 7^2 = 4 \cdot 49 = 196\) So, the third term of the geometric sequence is 196. The general term of the geometric sequence is \(a_m = 4 \cdot 7^{m-1}\) and the third term is 196.

Key Concepts

nth term formula

The nth term formula is a crucial tool in understanding and working with geometric sequences. It allows you to find any term in a sequence without listing all the preceding ones. This formula is given by:

• \(a_n = a_1 \cdot r^{n-1}\)

Here, \(a_n\) represents the nth term you want to find, \(a_1\) is the first term in the sequence, \(r\) is the common ratio, and \(n\) is the number of the term you are solving for.
For instance, to find the third term, you'd set \(n = 3\). This formula simplifies calculations and helps you understand the relationship between terms in a geometric sequence.

common ratio

The common ratio is an essential aspect of a geometric sequence. It describes the factor by which we multiply one term to get the next term.
In the given exercise, the common ratio is 7, indicated by \(r\).
The common ratio can be identified by dividing any term (except the first term) by the one before it, and it remains constant throughout the sequence.
Understanding the common ratio helps in predicting the pattern and values of the sequence effortlessly.

sequence calculation

Sequence calculation involves using the nth term formula to determine specific terms in a geometric sequence.
To calculate a term:

• First, you need to know the first term \(a_1\), the common ratio \(r\), and the term position \(n\).
• Then, substitute these values into the nth term formula: \(a_n = a_1 \cdot r^{n-1}\).
• Finally, evaluate the expression by performing the arithmetic operations.

For example, find the third term of a sequence with \(a_1 = 4\) and \(r = 7\) by setting \(n = 3\). Substitute these values in the formula:
\(a_3 = 4 \cdot 7^{3-1} = 4 \cdot 49 = 196\).
This shows that the third term is 196, demonstrating how effective sequence calculation can be.

first term in sequence

The first term in a geometric sequence, denoted as \(a_1\), sets the starting point for all subsequent calculations in the sequence. It is the foundation upon which the entire sequence is built.
In our example, the first term is 4.
This value is multiplied by the common ratio raised to the power of the position minus one (as per the nth term formula) to find any other term in the sequence.
Understanding the role of the first term is critical because it determines the scale of the sequence and affects every term that follows.