Problem 35 - Precalculus Mathematics for Calculus

Question

Determine the common ratio, the fifth term, and the \(n\) th term of the geometric sequence. $$1, s^{2 / 7}, s^{4 / 7}, s^{6 / 7}, \dots$$

Step-by-Step Solution

Answer

Common ratio is \(s^{2/7}\); fifth term is \(s^{8/7}\); \(n\)-th term is \(s^{2(n-1)/7}\).

Step 1: Identify the Common Ratio

In a geometric sequence, each term is obtained by multiplying the previous term by a constant, called the common ratio. To find it, divide the second term by the first term.The first term is 1, and the second term is \(s^{2/7}\). Thus, the common ratio \(r\) is given by:\[r = \frac{s^{2/7}}{1} = s^{2/7}\]

Step 2: Find the Fifth Term

The general formula for the \(n\)-th term of a geometric sequence is \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio.Plug in the known values to find the fifth term:\[a_5 = 1 \cdot (s^{2/7})^{5-1} = s^{8/7}\]

Step 3: Derive the Formula for the nth Term

Using the formula for the \(n\)-th term \(a_n = a_1 \cdot r^{n-1}\), where \(a_1 = 1\) and \(r = s^{2/7}\), we can deduce:\[a_n = 1 \cdot (s^{2/7})^{n-1} = s^{2(n-1)/7}\]This gives us the expression for the \(n\)-th term of the sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key element that defines the relationship between its consecutive terms. The common ratio remains constant throughout the sequence. Understanding the common ratio is essential for fully grasping how a geometric sequence progresses. To determine it, we divide the second term by the first term.

Let's consider the sequence provided: the first term is 1 and the second term is \(s^{2/7}\). By dividing these, we find:

• Common Ratio \(r\) = \(\frac{s^{2/7}}{1} = s^{2/7}\)

This common ratio of \(s^{2/7}\) tells us that each term in the sequence is created by multiplying the previous term by \(s^{2/7}\). This multiplication pattern is what makes the sequence a geometric one.

N-th Term

The nth term of a geometric sequence is critical because it provides a formulaic way to find any specific term in the sequence without writing out all preceding terms. The nth term formula answers the question: "What is the value of the term at position \(n\)?"

For our sequence, the formula is derived from the first term and the common ratio:

• The first term \(a_1\) is 1.
• The common ratio \(r\) is \(s^{2/7}\).
• The formula for the nth term is \(a_n = a_1 \cdot r^{n-1}\).

Substituting the known values yields:

• \(a_n = 1 \cdot (s^{2/7})^{n-1} = s^{2(n-1)/7}\)

This formula provides a shortcut to any term in the sequence by plugging in the value of \(n\), allowing us to bypass the step-by-step calculations.

Geometric Sequence Formula

The geometric sequence formula is an invaluable tool for analyzing sequences. It serves as a universal pattern that allows us to compute any term without sequential multiplication. This formula is:

• \(a_n = a_1 \cdot r^{n-1}\)

In this equation:

• \(a_n\) represents the nth term.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the term number in the sequence.

Using this formula, one can promptly determine any term's value by simply knowing the position \(n\), first term, and common ratio, illustrating the power of mathematical generalization.

Sequence Terms

Understanding the terms in a geometric sequence involves recognizing that each term is generated by a particular relationship: the multiplication of the previous term by the common ratio. This multiplication is repeated for each term, building up the sequence.

For the given sequence \(1, s^{2/7}, s^{4/7}, s^{6/7}, \ldots\), we start with an initial term:

• The first term \(a_1 = 1\)
• Subsequent terms: each is \(s^{2/7}\) times the previous one, such as \(s^{2/7}, s^{4/7}, s^{6/7}\), and so forth.

Understanding sequence terms helps in visualizing the pattern and form mathematical predictions or calculations for future terms using the common ratio and any starting point. By recognizing this structured growth, listeners and learners can apply the concept of geometric progress to various mathematical and real-world contexts.