Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Find \(a_{1}\) and \(r\) for each geometric sequence. $$a_{4}=-\frac{1}{4}, a_{9}=-\frac{1}{128}$$

Step-by-Step Solution

Answer

\(a_1 = -2, r = \frac{1}{2}\).

Step 1: Determine the formula for geometric sequences

In a geometric sequence, the n-th term is given by the formula \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio.

Step 2: Substitute terms into the formula for two equations

We have the terms \(a_4\) and \(a_9\) given. Using the formula \(a_n = a_1 \cdot r^{n-1}\), we can write two equations: \(a_1 \cdot r^3 = -\frac{1}{4}\) and \(a_1 \cdot r^8 = -\frac{1}{128}\).

Step 3: Divide the equations to solve for the common ratio

Divide the second equation by the first to eliminate \(a_1\), resulting in \(\frac{a_1 \cdot r^8}{a_1 \cdot r^3} = \frac{-\frac{1}{128}}{-\frac{1}{4}}\). This simplifies to \(r^5 = \frac{1}{32}\).

Step 4: Solve for the common ratio

Take the fifth root of each side: \(r = \left(\frac{1}{32}\right)^{1/5}\). Calculate \(\frac{1}{32}\) to the power of \(1/5\) to find \(r \approx \frac{1}{2}\).

Step 5: Substitute back to find the first term

Use the value of \(r\) to find \(a_1\) using one of the original equations. Substitute into \(a_1 \cdot \left(\frac{1}{2}\right)^3 = -\frac{1}{4}\) to compute \(a_1 = -\frac{1}{4} \cdot \frac{1}{8} = -2\).

Key Concepts

Understanding the Common Ratio in Geometric Sequences

In geometric sequences, the **common ratio** plays a vital role in determining the relationship between successive terms. It is the factor by which you multiply one term to obtain the next term. If you think of the sequence as a series of steps, the common ratio is the size of the step you take each time.

• To find the common ratio, divide any term by its preceding term. For instance, if your sequence is 2, 4, 8, 16, the common ratio is 4 divided by 2, which equals 2.
• The common ratio can be a fraction, whole number, or even a negative number, affecting the sequence by either shrinking, expanding, or flipping its terms respectively.

In our exercise, we solved for the common ratio by creating two equations from the given terms and simplifying. By dividing the equations, we canceled out other variables, isolating \(r^5 = \frac{1}{32}\), and then solving for \(r\), we found the common ratio to be \(\frac{1}{2}\). Remember, understanding this step is crucial because every term in a geometric sequence relies on it.

Finding the First Term in a Geometric Sequence

The **first term**, often denoted as \(a_1\), serves as the foundation or starting point of a geometric sequence. Knowing the first term is essential because it is part of the formula used to calculate any term in the sequence. In any geometric sequence problem, identifying \(a_1\) is like establishing the starting point of your journey.

• Once the common ratio is known, \(a_1\) can often be found by substituting known values back into one of the original equations derived from the terms given in the sequence.
• In the context of our exercise, once \(r\) was determined as \(\frac{1}{2}\), we used the term \(a_4 = -\frac{1}{4}\) to find \(a_1\). By rearranging the equation \(a_1 \cdot \left(\frac{1}{2}\right)^3 = -\frac{1}{4}\) and solving, we discovered that \(a_1 = -2\).

This initial term \(a_1 = -2\) kickstarts our sequence, giving us the definitive beginning from which all other terms arise through multiplication by the common ratio.

Finding the N-th Term in a Geometric Sequence

The **n-th term** of a geometric sequence lets you find any term in the sequence without listing all preceding ones. This ability is particularly useful for finding terms far along in the sequence without navigating through all prior terms. To find the \(n\)-th term, you use the formula:\[ a_n = a_1 \cdot r^{n-1} \]Here’s how it works:

• Identify \(a_1\), the first term of your sequence. In our scenario, \(a_1\) was determined to be \(-2\).
• Use \(r\), your common ratio, which we found to be \(\frac{1}{2}\) in the original problem.
• To find the \(n\)-th term, plug \(n\) into the formula. For example, for the 10th term, substitute \(n=10\) into the formula.

This formula is highly versatile and can be utilized for any \(n\)-th term you need to find, giving you a powerful tool to explore sequences without hassle. It shows how the sequence extends infinitely, governed by the mechanisms established by \(a_1\) and \(r\). Remember, by combining the first term and the common ratio, this formula opens the full potential of exploring any term in the sequence.