Problem 47 - A Graphical Approach to College Algebra

Question

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$12,24,48,96, \dots$$

Step-by-Step Solution

Answer

The series does not converge because \(r = 2 > 1\).

Step 1: Identify first term

The first term (also known as the common starting point) of the geometric sequence is identified as 12. This is denoted as \(a = 12\).

Step 2: Identify the second term

The second term of the sequence is given as 24. This value will be used to determine the common ratio \(r\).

Step 3: Calculate the common ratio

The common ratio \(r\) is calculated by dividing the second term by the first term, i.e. \(r = \frac{24}{12} = 2\).

Step 4: Determine if the series converges

For a geometric series to converge, the absolute value of the common ratio \(|r|\) must be less than 1. Here, \(|r| = 2| > 1\).

Step 5: Conclusion

Since \(|r| > 1\), this geometric series does not converge.

Key Concepts

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply a term to get the next one. To find this ratio, take any term in the sequence (after the first one), and divide it by the term preceding it.
For example, given the sequence 12, 24, 48, 96, ..., we can determine the common ratio, denoted as \( r \), by dividing the second term by the first. So, \( r = \frac{24}{12} = 2 \).
This consistent ratio helps identify the pattern and progression of the sequence. It answers the question: "By what factor do the terms increase or decrease?"
Every term in this sequence is derived from the first term by multiplying it repeatedly by \( r \). However, when \( |r| \)> 1, subsequent terms can grow large enough to make the series diverge.

Geometric Series Convergence

An infinite geometric series converges when the sum of its terms approaches a definite value as more and more terms are added. The key factor determining this is the absolute value of the common ratio \( |r| \).
For convergence, \( |r| \) must be less than 1. This means the terms steadily decrease in magnitude, eventually totaling a finite number. If \( |r| \geq 1 \), as in the sequence given, the series diverges.
In the original example, the ratio is 2, which means:

• Terms are growing larger.
• The series diverges, as it will not sum to a finite value.

When the terms don’t approach a fixed sum, such a series is considered to be diverging.

First Term

The first term of a geometric sequence acts as the building block for the entire series. It is usually denoted by \( a \) and serves as the starting point from which all other terms are derived through multiplication by the common ratio.
In the sequence 12, 24, 48, 96, ..., the first term is 12, noted as \( a = 12 \).
This first term is crucial because:

• It serves as the base value.
• It influences the size of subsequent terms.
• In conjunction with \( r \), it determines the structure of the sequence.

Without it, the sequence would not be anchored, making it impossible to calculate other terms. Understanding the role of the first term is necessary for dissecting any geometric sequence.