Problem 49 - A Graphical Approach to College Algebra

Question

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$-48,-24,-12,-6, \dots$$

Step-by-Step Solution

Answer

The common ratio \( r \) is \( \frac{1}{2} \), and the sequence converges.

Step 1: Identify the first term and the common ratio

The first term of the sequence, \( a \), is given as \( -48 \). The common ratio, \( r \), can be found by dividing the second term by the first term. Thus, \( r = \frac{-24}{-48} = \frac{1}{2} \).

Step 2: Determine if the series converges

An infinite geometric series converges if the absolute value of the common ratio, \( |r| \), is less than 1. Here, \( |\frac{1}{2}| = 0.5 \), which is less than 1. Therefore, the series converges.

Step 3: Check for other terms and their progression

Continue checking the sequence to ensure that the same common ratio is maintained across additional terms. Dividing \( -12 \) by \( -24 \) gives \( \frac{1}{2} \) and \( -6 \) by \( -12 \) also gives \( \frac{1}{2} \), confirming consistency.

Key Concepts

Geometric Series Convergence

When dealing with the concept of geometric series, especially infinite ones, the notion of convergence is key. An infinite geometric series converges only if its common ratio, denoted as \( r \), satisfies the condition \( |r| < 1 \). This means that the terms of the series get indefinitely smaller, approaching zero, as you move along the sequence.

• If this condition is met, you can confidently state that the series has a sum that approaches a finite limit.
• However, if \( |r| \geq 1 \), the series diverges, meaning the sum does not stabilize at a particular value and increases without bound.

In the given example, since the calculated \( |r| = 0.5 \), the series is indeed convergent. Knowing whether a series converges tells mathematicians a lot about its behavior and potential applications.

Common Ratio Calculation

The common ratio in a geometric sequence is a pivotal element used to elucidate the pattern of the sequence. Calculating it is straightforward: divide any term in the sequence by the previous term.
For instance, using the sequence provided:

• The second term is \(-24\), while the first term is \(-48\).
• To find the common ratio \( r \), compute \( r = \frac{-24}{-48} = \frac{1}{2} \).

This process is repeated for additional terms if needed, to ensure consistency throughout the sequence.

• For example, dividing \(-12\) by \(-24\) also gives \( \frac{1}{2} \), as does \(-6\) by \(-12\).

By confirming the same common ratio with multiple terms, one can ascertain the uniformity of the sequence, which is essential for making conclusions about the convergence of the series.

Mathematical Sequences

Sequences in mathematics are ordered sets of numbers following a specific rule or pattern. Geometric sequences, in particular, have terms that form a constant ratio.

• Each term is generated by multiplying the previous term by the common ratio \( r \).
• They are expressed as \( a, ar, ar^2, ar^3, \ldots \), where \( a \) is the first term and \( r \) is the common ratio.

Understanding how these sequences behave is critical for solving problems involving infinite series and their sums. Different types of sequences can model various natural and mathematical phenomena, making them essential tools in numerous fields.
When studying sequences, it's also important to understand how changes in the common ratio or initial term can impact the entire sequence or its behavior as a series, convergent or otherwise.