Problem 61 - Algebra 2

Question

Write the explicit formula for each geometric sequence. Then generate the first three terms. $$ a_{1}=3, r=\frac{3}{2} $$

Step-by-Step Solution

Answer

The explicit formula for the geometric sequence is \(a_n = 3 * (\frac{3}{2})^{(n-1)}\). The first three terms of the sequence are 3, \(\frac{9}{2}\), \(\frac{27}{4}\).

Step 1: Write the Explicit Formula

Using the given first term and the common ratio, plug-in the values into the explicit formula of geometric sequence, i.e. \(a_n = a_1 * r^{(n-1)}\). So, the explicit formula for this sequence is \(a_n = 3 * (\frac{3}{2})^{(n-1)}\)

Step 2: Generate First Term

To generate the first term, substitute \(n = 1\) into the explicit formula \(a_n = 3 * (\frac{3}{2})^{(1-1)} = 3 * 1 = 3\)

Step 3: Generate Second Term

To generate the second term, substitute \(n = 2\) into the explicit formula \(a_n = 3 * (\frac{3}{2})^{(2-1)} = 3 * \frac{3}{2} = \frac{9}{2}\)

Step 4: Generate Third Term

To generate the third term, substitute \(n = 3\) into the explicit formula \(a_n = 3 * (\frac{3}{2})^{(3-1)} = 3 * (\frac{3}{2})^2 = 3 * \frac{9}{4} = \frac{27}{4}\)

Key Concepts

Explicit Formula

An explicit formula is a mathematical expression used to find any term in a sequence directly. For a geometric sequence, the explicit formula is given by:

• \( a_n = a_1 \cdot r^{(n-1)} \)

where:

• \( a_n \) is the \( n^{th} \) term in the sequence,
• \( a_1 \) is the first term,
• \( r \) is the common ratio, and
• \( n \) is the term number.

Plugging the values from the exercise, we get \( a_n = 3 \cdot \left(\frac{3}{2}\right)^{(n-1)} \). This formula helps us calculate any term in the sequence without needing to go through all previous terms. It saves time and is particularly useful for large sequences.
Just input \( n \) as whatever term number you need, and it gives you the exact term.

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get the next term. In the given example, the common ratio \( r \) is \( \frac{3}{2} \). This means each term is obtained by multiplying the previous term by \( \frac{3}{2} \).
Understanding the common ratio is crucial because it defines the sequence's behavior:

• If \( r > 1 \): the sequence grows,
• If \( 0 < r < 1 \): the sequence decreases,
• If \( r = 1 \): the sequence is constant, and
• If \( r < 0 \): the sign of each term alternates.

The common ratio helps in forming the explicit formula by considering how each term is related to the first term.

First Term

The first term in a geometric sequence is where it all begins. It's represented by \( a_1 \). For the sequence in this exercise, \( a_1 = 3 \).
The first term is pivotal because:

• It acts as the base value in the explicit formula,
• It's the starting point for generating all sequence terms using the common ratio.

When multiplying the first term by the common ratio repeatedly, we generate all subsequent terms in the sequence. This term is crucial because it forms the foundation from which the entire sequence builds.

Sequence Terms

Sequence terms in a geometric sequence are the specific values found using the formula. Using the explicit formula:\[ a_n = 3 \cdot \left(\frac{3}{2}\right)^{(n-1)} \]
You can determine each term as follows:

• First term \( (n=1) \): \[ a_1 = 3 \]
• Second term \( (n=2) \): \[ a_2 = 3 \cdot \frac{3}{2} = \frac{9}{2} \]
• Third term \( (n=3) \): \[ a_3 = 3 \cdot \left(\frac{3}{2}\right)^2 = \frac{27}{4} \]

By substituting different values of \( n \) into the explicit formula, you can find any term in the sequence. Each term follows from the previous through multiplication by the common ratio. This pattern makes the sequence predictable and easy to navigate.