Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Find all possible values of \(r\) for a geometric sequence with the two given terms. $$a_{3}=4, a_{7}=\frac{1}{4}$$

Step-by-Step Solution

Answer

The possible values for \(r\) are \( \frac{\sqrt{2}}{2} \) and \( -\frac{\sqrt{2}}{2} \).

Step 1: Express Terms in Terms of First Term and Common Ratio

In a geometric sequence, any term can be expressed as a multiple of the first term and the common ratio raised to an appropriate power. Therefore, we can express the terms as follows:\[ a_3 = ar^2 = 4 \]\[ a_7 = ar^6 = \frac{1}{4} \]

Step 2: Divide the Equations to Remove First Term Dependence

To find the common ratio \(r\), we divide the second equation by the first to eliminate \(a\):\[ \frac{a_7}{a_3} = \frac{ar^6}{ar^2} = r^4 = \frac{1}{4} \]

Step 3: Solve for the Common Ratio

Solve the equation \(r^4 = \frac{1}{4}\) by taking the fourth root of both sides:\[ r = \sqrt[4]{\frac{1}{4}} = \frac{1}{\sqrt{2}} \text{ or } r = -\frac{1}{\sqrt{2}} \]We have two possible values for \(r\), positive or negative, since both will satisfy \(r^4 = \frac{1}{4}\).

Step 4: Simplify the Common Ratio

Simplify \( \frac{1}{\sqrt{2}} \) by multiplying the numerator and denominator by \( \sqrt{2} \), so we have:\[ r = \frac{\sqrt{2}}{2} \text{ or } r = -\frac{\sqrt{2}}{2} \]

Key Concepts

Common Ratio

In a geometric sequence, each term is obtained by multiplying the previous term by a constant known as the common ratio, denoted as \( r \). This ratio is crucial because it determines the behavior of the entire sequence.
The common ratio can be:

• Positive or negative: This impacts whether the sequence alternates signs or remains consistent.
• Greater than 1: This results in an increasing sequence.
• Between 0 and 1: This results in a decreasing sequence.
• Negative: This causes the sequence to alternate between positive and negative values.

To find the common ratio:1. Divide any term by the previous term: \( r = \frac{a_{n+1}}{a_n} \).2. Use known terms of the sequence and solve for \( r \) when expressions are provided, as in our example: \( \frac{a_7}{a_3} = r^4 = \frac{1}{4} \), leading to \( r = \pm \frac{\sqrt{2}}{2} \).

Terms of a Sequence

The terms of a geometric sequence are generated by applying the common ratio successively. Understanding how these terms relate, especially when some are given, helps in solving related problems.

In our example, we had the terms:

• \( a_3 = 4 \), representing the third term of the sequence.
• \( a_7 = \frac{1}{4} \), which is the seventh term of the sequence.

Each term \( a_n \) can be expressed as \( a \, r^{n-1} \), where \( a \) is the first term and \( n \) is the position in the sequence. Knowing this formula allows for calculations involving any term:

• For the third term: \( a_3 = ar^2 = 4 \).
• For the seventh term: \( a_7 = ar^6 = \frac{1}{4} \).

Setting up equations based on known terms helps deduce other sequence properties, like the first term or common ratio.

Solving Equations

Equations involving geometric sequences often require manipulation to isolate desired variables, such as the common ratio or specific terms.

In the example,

• We started with the equations \( a_3 = ar^2 \) and \( a_7 = ar^6 \).
• We divided these to isolate the common ratio, leading to \( \frac{a_7}{a_3} = r^4 \).

The goal in solving such equations is to simplify and solve for the unknown:

• Take roots to solve power equations, like \( r^4 = \frac{1}{4} \), by finding \( \sqrt[4]{\frac{1}{4}} \).
• Solutions, such as \( r = \pm \frac{\sqrt{2}}{2} \), show multiple possible values that satisfy the condition due to the properties of roots.

Simplification steps, like rationalizing denominators, can also be necessary to present results more acceptably in mathematical terms.