Problem 26 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=81, a_{5}=1 $$

Step-by-Step Solution

Answer

The first five terms are 81, 27, 9, 3, 1.

Step 1: Identify the general formula for a geometric sequence

A geometric sequence is defined by its first term, \(a_1\), and common ratio, \(r\). The general term, \(a_n\), of a geometric sequence can be written as \(a_n = a_1 \, r^{n-1}\).

Step 2: Set up equations for terms

We know \(a_1 = 81\) and \(a_5 = 1\). Using the formula \(a_n = a_1 \, r^{n-1}\), we set up the equation for the fifth term: \(a_5 = 81 \, r^{4} = 1\).

Step 3: Solve for the common ratio

We solve the equation from Step 2: \(81 \, r^{4} = 1\). So, \(r^{4} = \frac{1}{81}\). Since \(81 = 3^4\), \(r^{4} = (\frac{1}{3})^4\), thus \(r = \frac{1}{3}\).

Step 4: Write the first five terms

Now that we know \(r = \frac{1}{3}\), we can find the first five terms: \(a_1 = 81\), \(a_2 = 81 \, \frac{1}{3}\), \(a_3 = 81 \, (\frac{1}{3})^2\), \(a_4 = 81 \, (\frac{1}{3})^3\), \(a_5 = 81 \, (\frac{1}{3})^4 = 1\).

Step 5: Simplify each term

Calculate each term using \(r = \frac{1}{3}\):- \(a_1 = 81\)- \(a_2 = 27\)- \(a_3 = 9\)- \(a_4 = 3\)- \(a_5 = 1\)

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is crucial for determining how each term in the sequence relates to the previous one. Each term is the product of the previous term and this constant ratio. For instance, if the common ratio is 2, then each term is twice the previous term. Conversely, if the ratio is \( \frac{1}{2} \), each term is half of the previous term.
Understanding the common ratio helps us quickly construct a geometric sequence. You can find this ratio by dividing any term by its preceding term in the sequence. In the exercise above, given the first term \( a_1 = 81 \) and the fifth term \( a_5 = 1 \), we set up an equation \( 81 \, r^{4} = 1 \). Solving this yields \( r = \frac{1}{3} \), showing each term is a third of the previous term.

General Term Formula

The general term formula of a geometric sequence allows us to calculate any term if we know the first term and the common ratio. This formula is expressed as \( a_n = a_1 \, r^{n-1} \). Here, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the position of the term within the sequence.
This formula is invaluable for advancing from one term to any other in the sequence without computing intermediary steps. For instance, to find the fifth term in our example, we input \( a_1 = 81 \) and \( r = \frac{1}{3} \) into the formula, finding that \( a_5 = 81 \, \left(\frac{1}{3}\right)^4 \), which equals 1.

Sequence Terms

The sequence terms in a geometric sequence rely entirely on the first term and the common ratio. These terms unfold as a pattern from these two components.
Once you establish the common ratio, subsequent terms are computed by multiplying the previous term by \( r \). For our sequence:

• \( a_1 = 81 \)
• \( a_2 = 81 \times \frac{1}{3} = 27 \)
• \( a_3 = 27 \times \frac{1}{3} = 9 \)
• \( a_4 = 9 \times \frac{1}{3} = 3 \)
• \( a_5 = 3 \times \frac{1}{3} = 1 \)

This sequence exemplifies a consistent pattern defined by multiplying by its common ratio, \( \frac{1}{3} \).

Equation Solving

Solving equations in geometric sequences often involves finding unknown elements like the common ratio. This might require unraveling equations based on the general term formula.
In our example, the equation \( 81 \, r^{4} = 1 \) represents how the first and fifth terms are connected through the ratio. We solved for \( r \) by isolating it: first expressing \( r^{4} = \frac{1}{81} \). Recognizing \( 81 \) as \( 3^4 \), it follows that \( r = \frac{1}{3} \).
Solving these equations efficiently often involves recognizing powers of numbers and employing properties of exponents.

First Term

The first term of a geometric sequence serves as its starting point and is fundamental for using the general term formula. Knowing this initial value enables us to compute the entire sequence once the common ratio is established.
In the exercise, the first term is \( a_1 = 81 \). Combined with the common ratio \( r = \frac{1}{3} \), the first term allows us to derive all following terms using the formula \( a_n = a_1 \, r^{n-1} \).
This starting point anchors the sequence and, along with the common ratio, unlocks the entire pattern of a geometric sequence.