Problem 23 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=1, a_{3}=16 $$

Step-by-Step Solution

Answer

The first five terms are \(1, 4, 16, 64, 256\) for \(r=4\), and \(1, -4, 16, -64, 256\) for \(r=-4\).

Step 1: Understanding 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 as \(r\). The sequence is expressed as \(a_1, a_2, a_3, \ldots\).

Step 2: Setting Up the Equation for Terms

Given \(a_1 = 1\) and \(a_3 = 16\), using the formula for a geometric sequence \(a_n = a_1 \cdot r^{n-1}\), we write \(a_3 = a_1 \cdot r^2\). Therefore, \(16 = 1 \cdot r^2\).

Step 3: Solving for Common Ratio (\(r\))

By the equation \(16 = r^2\), solve for \(r\):\[r^2 = 16\]Therefore, \(r = \pm 4\).

Step 4: Constructing the Sequence with \(r=4\)

Using \(r = 4\), compute the first five terms:\(a_1 = 1\)\(a_2 = 1 \cdot 4 = 4\)\(a_3 = 4 \cdot 4 = 16\)\(a_4 = 16 \cdot 4 = 64\)\(a_5 = 64 \cdot 4 = 256\)

Step 5: Constructing the Sequence with \(r=-4\)

Using \(r = -4\), compute the first five terms:\(a_1 = 1\)\(a_2 = 1 \cdot (-4) = -4\)\(a_3 = -4 \cdot (-4) = 16\)\(a_4 = 16 \cdot (-4) = -64\)\(a_5 = -64 \cdot (-4) = 256\)

Key Concepts

Common Ratio

In any geometric sequence, understanding the 'common ratio' is essential. It is the constant factor by which one term is multiplied to get the next term. This ratio is denoted by the symbol \(r\). The common ratio plays a crucial role in determining the behavior of the sequence:

• If \(r > 1\), the sequence will exhibit exponential growth, meaning the terms will get larger and larger as you proceed.
• If \(0 < r < 1\), the sequence will decrease towards zero.
• If \(r = -1\), the sequence will alternate between values, creating a zig-zag pattern.
• If \(|r| > 1\), the terms will oscillate in size.

Knowing the common ratio allows us to find any term in the sequence using the formula for any nth term: \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term.

Sequence Terms

The sequence terms in a geometric progression are the individual numbers that make up the sequence. Each term is calculated by multiplying the previous term by the common ratio \(r\). When trying to find the first few terms of a geometric sequence:

• Start with the initial term \(a_1\).
• Multiply \(a_1\) by \(r\) to find \(a_2\).
• Continue this multiplication process to find successive terms.

In the given exercise, the first term \(a_1 = 1\) and the third term \(a_3 = 16\) were given, which led to solving for the common ratio \(r\). Using \(r = 4\), we obtained the sequence: \(1, 4, 16, 64, 256\). When \(r = -4\), the sequence became: \(1, -4, 16, -64, 256\). The change in sign of the common ratio results in an alternating sign of the terms.

Geometric Progression

A geometric progression, or geometric sequence, is a sequence of numbers where each term after the first is formed by multiplying the previous one by a fixed, non-zero number known as the common ratio. Here are a few characteristics and properties of geometric progressions:

• The ratio between successive terms is always the same, which defines the sequence's behavior.
• It can be represented generally as: \(a, ar, ar^2, ar^3, \ldots\)
• The nth term of a geometric sequence can be found using the formula: \(a_n = a_1 \cdot r^{n-1}\), which helps in finding terms without listing all previous terms.

The ability to find the common ratio from just two known terms, as demonstrated in the exercise, showcases the utility of geometric progressions. They allow for efficient calculation of terms across a sequence and can model exponential growth or decay seen in various real-world scenarios, such as population growth, radioactive decay, and financial interest.