Problem 18 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=\frac{1}{4}, r=-2 $$

Step-by-Step Solution

Answer

The first five terms are \(\frac{1}{4}, -\frac{1}{2}, 1, -2, 4\).

Step 1: Understand 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 as \( r \). Our sequence starts with \( a_1 = \frac{1}{4} \) and the common ratio \( r = -2 \).

Step 2: Calculate the Second Term

To find the second term \( a_2 \), multiply the first term by the common ratio: \[ a_2 = a_1 \times r = \frac{1}{4} \times (-2) = -\frac{1}{2} \].

Step 3: Calculate the Third Term

To find the third term \( a_3 \), multiply the second term by the common ratio: \[ a_3 = a_2 \times r = -\frac{1}{2} \times (-2) = 1 \].

Step 4: Calculate the Fourth Term

To find the fourth term \( a_4 \), multiply the third term by the common ratio: \[ a_4 = a_3 \times r = 1 \times (-2) = -2 \].

Step 5: Calculate the Fifth Term

To find the fifth term \( a_5 \), multiply the fourth term by the common ratio: \[ a_5 = a_4 \times r = -2 \times (-2) = 4 \].

Key Concepts

Common Ratio in Geometric Sequences

In a geometric sequence, the common ratio, denoted as \( r \), is a fundamental component. This ratio is constant between consecutive terms of the sequence. To gain a full understanding, note that every time you move from one term to the next, you multiply it by this fixed number.

• The common ratio can be any non-zero number. It can be positive or negative, and may sometimes be a fraction.
• It dictates the nature of the sequence, whether it increases, decreases, or oscillates between values.

For the sequence given, \( r = -2 \), meaning each term is the previous term multiplied by \(-2\). This negative ratio results in the sequence oscillating between positive and negative terms. Understanding the common ratio helps predict how the sequence will unfold and allows you to compute subsequent terms efficiently.

Terms of a Sequence

The terms of a sequence refer to the individual numbers that compose the entire sequence. Understanding each term and how it is derived is as important as understanding the sequence itself.

• The first term, \( a_1 \), is the starting point of the sequence. In our case, \( a_1 = \frac{1}{4} \).
• Each subsequent term is derived by multiplying the previous term with the common ratio.
• The terms follow a strict pattern as dictated by the ratio, allowing for predictability and easy calculation of further terms.

By clearly mapping out each step, such as moving from \( a_1 \) to \( a_2 \) and onwards, you can see how each term builds upon its predecessor. It not only reinforces the structure of the sequence but also highlights the power and simplicity of geometric progressions.

Multiplying Sequence Terms to Determine Next

In geometric sequences, the process of determining subsequent terms is built upon the foundational act of multiplication. Starting from the initial term, each next term is calculated by applying the common ratio.

• The operation involves simple multiplication of the current term with the common ratio \( r \).
• This multiplication is repeated iteratively to yield each sequential term.
• The predictability of this process allows for ease in calculating even distant terms without direct computation of each intermediary step.

Let's consider an example from the exercise: Starting with \( a_1 = \frac{1}{4} \), to find \( a_2 \), multiply by \(-2\) to get \(-\frac{1}{2}\). Continuing this operation, \( a_3 \) becomes \(1\), \( a_4 \) results in \(-2\), and \( a_5 \) gives \(4\). This systematic multiplication offers clarity and demonstrates the structured nature of geometric sequences.