Problem 18 - Algebra 2

Question

Find the first five terms of each geometric sequence described. $$ a_{1}=2, r=-3 $$

Step-by-Step Solution

Answer

2, -6, 18, -54, 162.

Step 1: Understanding the Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted by \( r \). In this exercise, the first term \( a_1 \) is \( 2 \) and the common ratio \( r \) is \( -3 \).

Step 2: Calculate the Second Term

The second term of the sequence \( a_2 \) is calculated by multiplying the first term \( a_1 \) by the common ratio \( r \): \[a_2 = a_1 imes r = 2 imes (-3) = -6.\]

Step 3: Calculate the Third Term

The third term \( a_3 \) is obtained by multiplying the second term \( a_2 \) by the common ratio \( r \): \[a_3 = a_2 imes r = -6 imes (-3) = 18.\]

Step 4: Calculate the Fourth Term

The fourth term \( a_4 \) is calculated by multiplying the third term \( a_3 \) by the common ratio \( r \): \[a_4 = a_3 imes r = 18 imes (-3) = -54.\]

Step 5: Calculate the Fifth Term

The fifth term \( a_5 \) is found by multiplying the fourth term \( a_4 \) by the common ratio \( r \): \[a_5 = a_4 imes r = -54 imes (-3) = 162.\]

Step 6: List the First Five Terms

Based on the calculations, the first five terms of the geometric sequence are: \( 2, -6, 18, -54, 162. \)

Key Concepts

Common Ratio

A common ratio is a crucial component of a geometric sequence. It is the constant factor by which each term in the sequence is multiplied to produce the next term. In our exercise, the common ratio is represented by \( r = -3 \). This means that every term in the sequence is obtained by multiplying the preceding term by \(-3\).

Understanding the common ratio is vital because it helps define the pattern and behavior of the sequence. If \( r \) is positive, the terms will all have the same sign. However, a negative \( r \), like \(-3\) in this exercise, leads to alternating signs in the sequence terms. Here's a simple illustration:

• If \( r = 3 \) and the first term is \( 2 \), the sequence will be \( 2, 6, 18, ... \).
• If \( r = -3 \) and the first term is \( 2 \), the sequence will be \( 2, -6, 18, ... \).

These examples show how the sign and magnitude of \( r \) impact the sequence. Consequently, knowing the common ratio is key to predicting the sequence terms.

Sequence Terms

In the context of a geometric sequence, the sequence terms are the numbers that you get by continuously applying the common ratio to the initial term. In our specific example, we start with the first term \( a_1 = 2 \).

Once we have the first term and the common ratio, we can determine further terms as follows:

• The second term, \( a_2 \), is found by multiplying \( a_1 \) by the common ratio, giving us \( 2 \times -3 = -6 \).
• The third term, \( a_3 \), is the product of \( a_2 \) and the common ratio, resulting in \( -6 \times -3 = 18 \).
• This pattern continues, and the sequence develops as \( 2, -6, 18, -54, 162 \), as shown in the step-by-step solution.

The sequence terms not only define the sequence itself but also help in understanding the exponential rate at which the sequence progresses due to the repeated multiplication by the common ratio.

Multiplication in Sequences

Multiplication plays a fundamental role in forming geometric sequences. It is the operation that binds the sequence together, applying the common ratio to transform each term into the next.

Each multiplication activity in the sequence serves a specific purpose:

• It ensures each term is generated in a consistent pattern, aligned with the structure given by the common ratio.
• For the given problem, each term is a direct result of multiplying the previous term by the common ratio \(-3\). This is evident in the calculations:
  • \( a_2 = 2 \times (-3) = -6 \)
  • \( a_3 = -6 \times (-3) = 18 \)
  • \( a_4 = 18 \times (-3) = -54 \)
  • \( a_5 = -54 \times (-3) = 162 \)

Such use of multiplication not only establishes the sequence but also highlights the power of geometric growth and decline based on the sign and size of the common ratio. It's a straightforward yet powerful concept illustrating exponential growth in mathematics.