Problem 33 - Precalculus: Mathematics for Calculus

Question

Find the \(n\)th term of a sequence whose first several terms are given. \(5,-25,125,-625, \dots\)

Step-by-Step Solution

Answer

The nth term is \( a_n = 5 \times (-5)^{n-1} \).

Step 1: Identify the Pattern

Observe the given sequence: 5, -25, 125, -625, ...Notice that each term is the previous term multiplied by -5. So the sequence is geometric with a common ratio, \( r = -5 \).

Step 2: Establish the Formula

For a geometric sequence, the general formula for the \(n\)th term \(a_n\) is given by:\[ a_n = a_1 \times r^{n-1} \]where \(a_1\) is the first term and \(r\) is the common ratio.

Step 3: Substitute Values

Substitute the known values into the formula. Here, \(a_1 = 5\) and \(r = -5\).Thus, the formula becomes:\[ a_n = 5 \times (-5)^{n-1} \]

Step 4: Simplify the Expression

The simplified expression for the \(n\)th term is:\[ a_n = 5 \times (-5)^{n-1} \]This expression represents the \(n\)th term of the sequence.

Key Concepts

Understanding the nth Term Formula in Geometric Sequences

In mathematics, finding the nth term of a sequence is a common task, especially useful when working with patterns or series. The nth term formula in a geometric sequence allows us to quickly find the specific term in the sequence using its position, without listing all previous terms. This is particularly efficient for longer sequences.

For a geometric sequence, the nth term formula is given by:

• \( a_n = a_1 \times r^{n-1} \)

- Here, \( a_n \) represents the nth term, which is the term at position n in the sequence.

- \( a_1 \) is the first term of the sequence.

- \( r \) is the common ratio, or the factor that each term is multiplied by to get to the next term.

This formula simplifies calculations by utilizing both the initial term and the consistent ratio, providing a straightforward method for predicting any term's value. Understanding how to apply this formula helps solve numerous sequence-related problems with ease.

Exploring the Common Ratio in Geometric Sequences

The term 'common ratio' refers to the consistent factor by which each number in a geometric sequence is multiplied to obtain the next number. Identifying the common ratio is essential for understanding the structure of a sequence and constructing its formula.

In the given sequence, 5, -25, 125, -625, ..., we observe the following pattern:

• To get from 5 to -25, we multiply by -5.
• From -25 to 125, we multiply by -5 again.
• This pattern continues throughout the sequence.

Therefore, the common ratio \( r \) is \(-5\). This value is vital as it directly influences the growth or decay, as well as the direction (positive or negative), of the sequence.

The common ratio helps predict subsequent terms and complements the nth term formula by forming the backbone of how the sequence progresses.

Utilizing the Geometric Sequence Formula

Geometric sequences are defined by their consistent structure, and the geometric sequence formula captures this by integrating the common ratio and the first term into a predictive model. The formula gives us a comprehensive way to explore the sequence's behavior across all terms.

The formula in question is:

• \( a_n = a_1 \times r^{n-1} \)

To utilize this:

• Identify the first term, \( a_1 \). In our example, it is 5.
• Determine the common ratio, \( r \), which is \(-5\) for our sequence.
• Plug these values into the formula to find any term you need. For instance, the 3rd term is \( 5 \times (-5)^{3-1} = 125 \).

Each component of this formula plays a critical role:

• \( a_1 \) sets the starting point.
• \( r^{n-1} \) calculates the multiplicative steps required to reach the nth term.

Using the geometric sequence formula, we gain a powerful tool for analyzing patterns, making it indispensable in sequences and series calculations.