Problem 28 - Algebra 2 and Trigonometry

Question

What is the 9 th term of the geometric sequence \(125,25,5, \ldots ?\)

Step-by-Step Solution

Answer

The 9th term of the sequence is \( \frac{1}{3125} \).

Step 1: Identify the First Term

In the given geometric sequence, the first term \( a \) is 125.

Step 2: Determine the Common Ratio

The common ratio \( r \) in a geometric sequence is found by dividing the second term by the first term. Thus, \( r = \frac{25}{125} = \frac{1}{5} \).

Step 3: Use the Formula for the n-th Term

The formula for the \( n \)-th term of a geometric sequence is \( a_n = ar^{n-1} \). We need to find the 9th term, so \( n = 9 \).

Step 4: Substitute and Calculate

Substitute the values into the formula: \( a_9 = 125 \left( \frac{1}{5} \right)^{9-1} \). Simplify to get \( a_9 = 125 \left( \frac{1}{5} \right)^8 \).

Step 5: Simplify the Expression

Calculate \( \left( \frac{1}{5} \right)^8 = \frac{1}{390625} \). So, \( a_9 = 125 \times \frac{1}{390625} = \frac{125}{390625} \).

Step 6: Perform Final Calculations

Simplify the fraction: \( \frac{125}{390625} = \frac{1}{3125} \) by dividing the numerator and denominator by 125.

Key Concepts

n-th Term Formula

The n-th term formula is vital when working with geometric sequences. In these sequences, each term is produced by multiplying the previous one by a constant called the common ratio. The formula for finding the n-th term in a geometric sequence is \( a_n = ar^{n-1} \), where:

• \( a_n \) is the n-th term you're looking for.
• \( a \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n-1 \) accounts for the number of times you multiply by the ratio after the first term.

To find the 9th term, you substitute the known values into this formula, using the first term of 125 and the common ratio of \( \frac{1}{5} \). Understanding this formula helps to quickly determine any term in the geometric sequence without listing all the previous terms.

Common Ratio

The common ratio \( r \) in a geometric sequence determines how each term relates to its predecessor. It is constant for any two consecutive terms in the sequence. For example, to find \( r \) you divide the second term of the sequence by the first. In the sequence 125, 25, 5, ..., \( r \) is calculated as follows:

• Second term \( 25 \)
• First term \( 125 \)
• \( r = \frac{25}{125} = \frac{1}{5} \)

This tells us that each term is \( \frac{1}{5} \) of the previous term. Recognizing the common ratio is crucial as it allows you to apply the n-th term formula, which further helps in finding any term within the sequence rapidly.

Simplifying Fractions

Simplifying fractions is a key skill necessary for working with geometric sequences, especially when calculating terms and expressing them in their simplest form. It involves reducing fractions to their lowest terms by finding any common factors of the numerator and denominator.

• To simplify \( \frac{125}{390625} \), identify any common factors.
• Both numbers are divisible by 125 which simplifies the fraction to \( \frac{1}{3125} \).

Simplification makes fractions easier to understand and compare, particularly in mathematical computations or when needing to provide answers in a clean form. It's always a good practice to simplify your results as it gives a clearer view of the actual scale or value of the term you've calculated.