Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, write the first five terms of the geometric sequence. \(a_{1}=-486, \quad a_{n}=-\frac{1}{3} a_{n-1}\)

Step-by-Step Solution

Answer

The first five terms are: \(-486, 162, -54, 18, -6\).

Step 1: Identifying the First Term

The problem states that the first term of the geometric sequence is \( a_1 = -486 \). This means that our sequence starts with \( -486 \).

Step 2: Understanding the Common Ratio

In the given sequence, each term is \( -\frac{1}{3} \) times the previous term. This tells us that the common ratio \( r = -\frac{1}{3} \).

Step 3: Finding the Second Term

To find the second term \( a_2 \), we multiply the first term by the common ratio: \( a_2 = -\frac{1}{3} \times (-486) = 162 \).

Step 4: Finding the Third Term

To find the third term \( a_3 \), we multiply the second term by the common ratio: \( a_3 = -\frac{1}{3} \times 162 = -54 \).

Step 5: Finding the Fourth Term

To find the fourth term \( a_4 \), we multiply the third term by the common ratio: \( a_4 = -\frac{1}{3} \times (-54) = 18 \).

Step 6: Finding the Fifth Term

To find the fifth term \( a_5 \), we multiply the fourth term by the common ratio: \( a_5 = -\frac{1}{3} \times 18 = -6 \).

Key Concepts

Understanding the First Term

In a geometric sequence, the first term is the starting point of the sequence. For our example, this means that the sequence begins with \(-486\).
Understanding the first term is important because it sets the initial value from which all future terms are calculated.
Every subsequent term in the sequence relies on this original starting point, which helps define the overall pattern of the sequence. When given a first term like \(-486\), it indicates that every calculated term after it originates directly from this number. By fixing this initial value, you can start to explore the relationship between each term in the sequence.

Exploring the Common Ratio

The common ratio in a geometric sequence is the constant value by which each term is multiplied to get to the next term.
For our sequence, the common ratio is \(-\frac{1}{3}\).
This tells us how each term relates to the previous one. When the ratio is applied, you can see that it consistently transforms each term into the next, creating a consistent pattern.

• If the common ratio is positive, the sequence will all have terms of the same sign.
• If the common ratio is negative, like our \(-\frac{1}{3}\), the signs of each term alternate.

In this case, every term after the first will be \(-\frac{1}{3}\) times the previous term, guiding the sequence along a specific path.

Terms of a Sequence Uncovered

The terms of a sequence are the individual elements that make up the pattern of the sequence.
In a geometric sequence, terms are determined by multiplying the previous term by the common ratio.

• The first term is already known. For our example, it’s \(-486\).
• From the first term, using a common ratio of \(-\frac{1}{3}\), you calculate the subsequent terms:


• Second term: \(162\)
• Third term: \(-54\)
• Fourth term: \(18\)
• Fifth term: \(-6\)

Each calculated term is part of the overall sequence, showing the dynamic interaction between the first term and the common ratio through repeated multiplication.

Cracking the Code with Algebra

Algebra helps us express the general form of any geometric sequence, providing a formula to calculate terms without having to multiply step-by-step.
For a sequence, where you know the first term and the common ratio, the nth term can be found using the formula: \[ a_n = a_1 \times r^{(n-1)} \] Given our problem: \( a_1 = -486 \) and \( r = -\frac{1}{3} \).
Using this algebraic representation, each term of the sequence can be derived quickly:

• 2nd term: \( -486 \times \left( -\frac{1}{3} \right)^1 \)
• 3rd term: \( -486 \times \left( -\frac{1}{3} \right)^2 \)
• 4th term: \( -486 \times \left( -\frac{1}{3} \right)^3 \)
• 5th term: \( -486 \times \left( -\frac{1}{3} \right)^4 \)

Algebra simplifies the process and allows for a wide range of calculations, making it easier to handle sequences and explore further possibilities.