Problem 31 - Precalculus: Mathematical for Calculus

Question

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

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

The given sequence is a geometric sequence: \(1, s^{2/7}, s^{4/7}, s^{6/7}, \ldots\). In a geometric sequence, each term is obtained by multiplying the previous term by a constant, known as the common ratio. The common ratio \(r\) can be found by dividing the second term by the first term: \(r = \frac{s^{2/7}}{1} = s^{2/7}\).

Step 2: Determine the Fifth Term

To find the fifth term of the sequence, we use the formula for the \(n\)-th term of a geometric sequence: \(a_n = a_1 \cdot r^{(n-1)}\). Here, \(a_1 = 1\), \(r = s^{2/7}\), and \(n = 5\). Substitute these values into the formula: \[a_5 = 1 \cdot (s^{2/7})^{4} = s^{8/7}.\] Thus, the fifth term of the sequence is \(s^{8/7}\).

Step 3: Find the General nth Term

The formula for the \(n\)-th term of a geometric sequence is \(a_n = a_1 \cdot r^{(n-1)}\). Substitute \(a_1 = 1\) and \(r = s^{2/7}\): \[a_n = 1 \cdot (s^{2/7})^{(n-1)} = s^{2(n-1)/7}.\] Hence, the general formula for the \(n\)-th term is \(s^{2(n-1)/7}\).

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is the factor by which we multiply each term to get to the next term. It's the magic number that helps to define the entire sequence. For the sequence given: \(1, s^{2/7}, s^{4/7}, s^{6/7}, \ldots \), determining the common ratio involves a simple division. We take the second term and divide it by the first term. Here, that means dividing \( s^{2/7} \) by \( 1 \), which gives us the common ratio \( r = s^{2/7} \).

• This constant ratio allows you to easily find other terms in the sequence.
• Every term is just the previous term multiplied by \( s^{2/7} \).
• Knowing the common ratio simplifies understanding of the sequence's behavior.

Using the nth Term Formula

The \( n \)-th term formula in a geometric sequence is essential for finding any term in the sequence without listing all previous ones. The formula is \( a_n = a_1 \cdot r^{(n-1)} \).

• \( a_1 \) is the first term of the sequence, which in our case is 1.
• \( r \) is the common ratio, which we've already determined as \( s^{2/7} \).
• \( n \) represents the position of the term in the sequence.

When we substitute into the formula for the \( n \)-th term of our sequence, we get \( a_n = 1 \cdot (s^{2/7})^{(n-1)} = s^{2(n-1)/7} \).
This compact expression lets you calculate any term in the sequence efficiently.

Determining the Fifth Term

Calculating specific terms in a geometric sequence leverages our earlier conclusions about the common ratio and the nth term formula. Finding the fifth term involves plugging values into the nth term formula.
Here, \( a_1 = 1 \), \( r = s^{2/7} \), and \( n = 5 \). We use these in the formula: \( a_5 = 1 \cdot (s^{2/7})^{4} \). This results in \( a_5 = s^{8/7} \).

• This specific calculation shows how using the nth term formula can quickly lead us to individual terms without repetition.
• The process can be applied to find any term where the position \( n \) is known.

The method is simple yet powerful for understanding geometric sequences.