Problem 35 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Common ratio: \(s^{2/7}\), Fifth term: \(s^{8/7}\), nth term: \(s^{2(n-1)/7}\).

Step 1: Identify the Geometric Sequence Elements

The given sequence is \(1, s^{2/7}, s^{4/7}, s^{6/7}, \ldots\). We are to determine the common ratio, the fifth term, and the general nth term of the sequence.

Step 2: Calculate the Common Ratio

A geometric sequence has the form \(a, ar, ar^2, ar^3, \ldots\), where \(a\) is the first term and \(r\) is the common ratio. Here, the first term \(a = 1\). The second term is \(s^{2/7}\). The common ratio \(r\) can be found using the formula: \[ r = \frac{s^{2/7}}{1} = s^{2/7} \]. Hence, the common ratio \(r = s^{2/7}\).

Step 3: Find the Fifth Term using the Common Ratio

To find the fifth term, we use the formula for the nth term of a geometric sequence: \[ a_n = ar^{n-1} \]. For the fifth term \(a_5\), it is \(a \times r^4\). So: \[ a_5 = 1 \times (s^{2/7})^4 = s^{8/7} \]. Therefore, the fifth term is \(s^{8/7}\).

Step 4: Determine the General nth Term

The general formula for the nth term of a geometric sequence is \(a_n = ar^{n-1}\). Here, \(a = 1\) and \(r = s^{2/7}\). Substituting these in gives: \[ a_n = 1 \times (s^{2/7})^{n-1} = s^{2(n-1)/7} \]. This is the formula for the nth term of the sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key feature that determines the pattern of the sequence. It is the factor by which each term in the sequence is multiplied to get the next term. Identifying the common ratio allows you to predict future terms and analyze the sequence's behavior.
The formula to find the common ratio \( r \) in a geometric sequence is:

• \( r = \frac{a_{n}}{a_{n-1}} \) where \( a_{n} \) is the current term and \( a_{n-1} \) is the previous term in the sequence.

In the example sequence \( 1, s^{2/7}, s^{4/7}, s^{6/7}, \ldots \), the first term \( a \) is \( 1 \) and the second term is \( s^{2/7} \).
Thus, the common ratio \( r \) is:

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

Once you know the common ratio, you can easily find any term in the sequence by multiplying the previous term by this ratio.

nth Term Formula

To find any term in a geometric sequence, it's essential to understand the nth term formula. This formula provides a shortcut to calculate any term directly without having to determine each preceding term.
The general form of the nth term \( a_n \) in a geometric sequence is given by:

• \( a_n = ar^{n-1} \)

Where:

• \( a \) is the first term.
• \( r \) is the common ratio.
• \( n \) is the term number.

Applying this formula to our sequence, where \( a = 1 \) and \( r = s^{2/7} \), you can find any term \( a_n \):

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

This formula is particularly useful as it allows for quick calculations of terms, saving time and reducing errors.

Sequences and Series

Sequences and series form the foundation of understanding patterns in mathematics. A sequence is a set of numbers arranged in a specific order following a specific rule. In contrast, a series is the sum of the terms of a sequence.
There are various types of sequences, with geometric sequences being one of them. In a geometric sequence, each term is derived by multiplying the previous term by a constant value known as the common ratio.
To summarize the essential points:

• A sequence is an ordered list of numbers.
• In a geometric sequence, the ratio between consecutive terms remains constant.
• While sequences deal with the arrangement of terms, series involve their summation.
• Understanding these concepts is crucial for solving complex mathematical problems and recognizing patterns in data.

Recognizing and playing with these structures enables you to understand more intricate mathematical ideas and real-world applications.