Problem 16 - A Graphical Approach to Precalculus with Limits

Question

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$3,-\frac{9}{4}, \frac{27}{16},-\frac{81}{64}, \dots$$

Step-by-Step Solution

Answer

\(a_5 = \frac{243}{256}\) and \(a_n = 3 \left( -\frac{3}{4} \right)^{n-1}\).

Step 1: Identify the First Term

The first term of the sequence, denoted by \( a_1 \), is \( 3 \). This is given as the first number in the sequence.

Step 2: Calculate the Common Ratio

The common ratio \( r \) of a geometric sequence can be found by dividing the second term by the first term. Therefore, \( r = \frac{-\frac{9}{4}}{3} = -\frac{3}{4} \).

Step 3: Find the Fifth Term

To find the fifth term \( a_5 \), use the formula for the \( n \)-th term in a geometric sequence: \( a_n = a_1 \cdot r^{n-1} \). Substitute \( a_1 = 3 \), \( r = -\frac{3}{4} \), and \( n = 5 \):\[ a_5 = 3 \left( -\frac{3}{4} \right)^{4} \].

Step 4: Simplify the Fifth Term Calculation

Calculate \( \left( -\frac{3}{4} \right)^{4} = \left( \frac{3}{4} \right)^{4} \) because raising to an even power removes the negative sign. This gives:\[ \left( \frac{3}{4} \right)^{4} = \frac{81}{256} \]. Now multiply by \( a_1 \):\[ a_5 = 3 \cdot \frac{81}{256} = \frac{243}{256} \].

Step 5: Find the General Term

The formula for the \( n \)-th term of a geometric sequence is given by: \( a_n = a_1 \cdot r^{n-1} \). Substitute \( a_1 = 3 \) and \( r = -\frac{3}{4} \):\[ a_n = 3 \left( -\frac{3}{4} \right)^{n-1} \].

Key Concepts

First Term Identification

In the realm of geometric sequences, one of the first steps is identifying the first term. This is crucial because it serves as the foundation from which the entire sequence is built. With the provided sequence, it starts off with the number 3. Hence, this is our first term and it is often denoted as \( a_1 \).

• The first term sets the initial value for your sequence calculations.
• It is always the starting point in the formula for any geometric sequence.

Understanding the first term helps students appreciate the importance of the beginning value in calculating subsequent terms.

Common Ratio Calculation

Once we know the first term, the next step is calculating the common ratio, denoted as \( r \). This ratio helps us understand how the sequence progresses from one term to the next. It is found by dividing the second term by the first term. In this sequence, we have:\[r = \frac{-\frac{9}{4}}{3} = -\frac{3}{4}.\]

• The common ratio indicates whether the sequence is increasing or decreasing, and by what factor.
• A negative value, like in this case, suggests that each term will alternate in sign.

Mastering the concept of a common ratio is essential for predicting the nature of a geometric sequence.

Nth Term Formula

Determining any term in a geometric sequence involves using the general formula for the nth term, \( a_n \). This formula is given by:\[a_n = a_1 \cdot r^{n-1}.\]By substituting \( a_1 = 3 \) and \( r = -\frac{3}{4} \), you can find any term in the sequence. For example, the fifth term is computed as follows:\[a_5 = 3 \cdot \left( -\frac{3}{4} \right)^{4}.\]

• This formula allows calculations of any term directly without listing all the intermediate terms.
• The exponent \( n-1 \) aligns the formula with the sequence, honoring the geometric pattern.

Consequently, mastering the nth term formula offers a powerful tool for understanding and utilizing geometric sequences.

Term Simplification

After determining the relevant term using the nth term formula, simplification is key. Simplifying the expression involves breaking down the powers and performing arithmetic operations clearly and precisely. For our sequence:First calculate \(( \left( -\frac{3}{4} \right)^{4} \), which simplifies to:\[\left( \frac{3}{4} \right)^{4} = \frac{81}{256},\]because raising a negative number to an even power results in a positive number.Next, we multiply this by the first term:\[a_5 = 3 \cdot \frac{81}{256} = \frac{243}{256}.\]

• Term simplification transforms complex expressions into practical results.
• This step is essential for handling longer sequences and practical applications where precision is needed.

Understanding how to simplify terms ensures clarity and correctness throughout mathematical calculations.