Problem 42 - Algebra 2

Question

Find the first five terms of each geometric sequence described. $$ a_{1}=243, r=\frac{1}{3} $$

Step-by-Step Solution

Answer

The first five terms are 243, 81, 27, 9, and 3.

Step 1: Understanding Geometric Sequences

In a geometric sequence, each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Here, the first term \( a_1 \) is 243 and the common ratio \( r \) is \( \frac{1}{3} \).

Step 2: Finding the Second Term

To find the second term \( a_2 \), multiply the first term by the common ratio: \( a_2 = a_1 \times r = 243 \times \frac{1}{3} = 81 \).

Step 3: Finding the Third Term

To find the third term \( a_3 \), multiply the second term by the common ratio: \( a_3 = a_2 \times r = 81 \times \frac{1}{3} = 27 \).

Step 4: Finding the Fourth Term

To find the fourth term \( a_4 \), multiply the third term by the common ratio: \( a_4 = a_3 \times r = 27 \times \frac{1}{3} = 9 \).

Step 5: Finding the Fifth Term

To find the fifth term \( a_5 \), multiply the fourth term by the common ratio: \( a_5 = a_4 \times r = 9 \times \frac{1}{3} = 3 \).

Key Concepts

Common Ratio

In the fascinating world of sequences, particularly geometric sequences, the term 'common ratio' is very significant. A geometric sequence is characterized by a sequence of numbers where each term after the first is found by multiplying the previous term by a constant. This constant is what we call the "common ratio." It remains the same throughout the sequence.

For example, if you have a starting term, let's say, 243, and your common ratio is \( \frac{1}{3} \), you use this common ratio to find all subsequent terms. The common ratio tells you how the sequence is changing. In our example, every term becomes one-third of the term before it, creating an interesting pattern of decreasing numbers.

Knowing the common ratio allows you to easily calculate any term in the sequence if you know the starting point, guiding you in manipulating and understanding the pattern and the relationships between terms.

Terms of a Sequence

In a geometric sequence, like the one described in the exercise, each 'term' represents a position in the sequence that is determined mathematically. The term number tells you where it is located within the sequence.

Each term is derived from multiplying the previous term by the common ratio. Using the given example, the first term \( a_1 \) is 243. From here, each subsequent term is calculated by multiplying the prior term by the common ratio \( \frac{1}{3} \).

• The second term is \( a_2 \ = 243 \times \frac{1}{3} = 81 \)
• The third term is \( a_3 \ = 81 \times \frac{1}{3} = 27 \)
• The fourth term is \( a_4 \ = 27 \times \frac{1}{3} = 9 \)
• The fifth term is \( a_5 \ = 9 \times \frac{1}{3} = 3 \)

Each term provides insight into how the sequence evolves when affected by the constant multiplication factor of the common ratio.

This process demonstrates that with the initial term and the common ratio, the entire sequence can be unraveled effortlessly.

Multiplication in Sequences

Understanding how multiplication plays a role in geometric sequences is key to mastering them. In such sequences, multiplication is the driving force that links one term to the next. It shapes the pattern and allows for determination of any term in the sequence.

In our example sequence starting at 243, each new term is derived by multiplying the previous one by \( \frac{1}{3} \). This repeated multiplication is fundamental because:

• It directs the sequence's growth (or in this case, reduction).
• It ensures the consistency and integrity of the sequence pattern.
• It easily generates any term when using the common ratio correctly.

This approach not only applies to number-based sequences but also in real-world scenarios where growth patterns or decay rates need to be understood, such as in population studies or financial forecasts.

Thus, seeing multiplication in sequences as a mechanism of continuity and change makes computation seamless and logically consistent.