Problem 41 - College Algebra

Question

For the following exercises, write an explicit formula for each geometric sequence. $$ a_{n}=\left\\{3,-1, \frac{1}{3},-\frac{1}{9}, \ldots\right\\} $$

Step-by-Step Solution

Answer

The explicit formula is \( a_n = 3 \left( -\frac{1}{3} \right)^{n-1} \).

Step 1: Identify the First Term

The first term of the sequence is given as \( a_1 = 3 \). This is the term that will be used as the starting point in our explicit formula for the sequence.

Step 2: Determine the Common Ratio

To find the common ratio \( r \) in the geometric sequence, we divide the second term by the first term. Thus, \( r = \frac{-1}{3} \). This consistent factor is what each term is multiplied by to get the next term in the sequence.

Step 3: Construct the Explicit Formula

We use the standard form for a geometric sequence explicit formula, which is \( a_n = a_1 \, r^{n-1} \). Substituting the values, we have:\[ a_n = 3 \left( -\frac{1}{3} \right)^{n-1} \].

Step 4: Simplify the Explicit Formula

The explicit formula simplified represents each term of the sequence based on its position \( n \). For \( n = 1, 2, 3, \ldots \), the formula \[ a_n = 3 \left( -\frac{1}{3} \right)^{n-1} \] generates the respective members of the sequence.

Key Concepts

Explicit Formula

An explicit formula in a geometric sequence allows us to find any term in the sequence without needing to calculate the previous terms. This is what makes it so powerful. For the sequence given

• The first term is 3.
• The common ratio is \( -\frac{1}{3} \).

The standard form of an explicit formula for a geometric sequence is represented as:

• \( a_n = a_1 \times r^{n-1} \)

where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. For our sequence, substituting in the known values gives us:

• \( a_n = 3 \times \left( -\frac{1}{3} \right)^{n-1} \)

This formula lets us find any term in the sequence easily. For example, when \( n = 3 \), we have \( a_3 = 3 \times \left( -\frac{1}{3} \right)^{2} \). This gives \( a_3 = \frac{1}{3} \), which matches the sequence.

Common Ratio

The common ratio is crucial for understanding geometric sequences. It is the factor that we multiply by each term to get to the next term. To find it, we divide a term in the sequence by the previous term. For our sequence:

• The second term is \(-1\).
• The first term is 3.

Using the formula \( r = \frac{a_2}{a_1} \), we calculate:

• \( r = \frac{-1}{3} \).

This common ratio \( r \) determines how the sequence progresses. Each term is

• multiplied by \( r \) to get the next term

which means if the first term is 3, the second term becomes \( 3 \times -\frac{1}{3} = -1 \). Knowing the common ratio helps us build the entire sequence and check the correctness of our explicit formula.

Sequence Terms

The terms of a geometric sequence are the individual elements that make up the sequence. Each term is represented as \( a_n \), where \( n \) is the position of the term. For our sequence:

• \( a_1 = 3 \)
• \( a_2 = -1 \)
• \( a_3 = \frac{1}{3} \)
• \( a_4 = -\frac{1}{9} \)

These terms follow a pattern defined by the explicit formula \( a_n = 3 \times \left( -\frac{1}{3} \right)^{n-1} \). Each term is important in identifying the properties of the sequence.

• The first term, \( a_1 \), sets the starting value.
• Subsequent terms are obtained by applying the common ratio \( r \).

Understanding the sequence terms and their relation through the formula can help in predicting future terms or identifying specific positions without manual calculation. For instance, knowing the sequence terms can make calculations easier when solving more complex problems related to geometric sequences.