Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio. \(a_{1}=8, \quad r=0.3\)

Step-by-Step Solution

Answer

The first five terms are 8, 2.4, 0.72, 0.216, 0.0648.

Step 1: Understanding the Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number, known as the common ratio. In this exercise, the first term \(a_1\) is given as 8, and the common ratio \(r\) is 0.3.

Step 2: Finding the Second Term

To find the second term \(a_2\) of the sequence, multiply the first term \(a_1 = 8\) by the common ratio \(r = 0.3\). So, \(a_2 = 8 \times 0.3 = 2.4\).

Step 3: Calculating the Third Term

The third term \(a_3\) is calculated by multiplying the second term \(a_2 = 2.4\) by the common ratio 0.3. Thus, \(a_3 = 2.4 \times 0.3 = 0.72\).

Step 4: Determining the Fourth Term

Next, find the fourth term \(a_4\) by multiplying the third term \(a_3 = 0.72\) by the common ratio 0.3. So, \(a_4 = 0.72 \times 0.3 = 0.216\).

Step 5: Finding the Fifth Term

Finally, compute the fifth term \(a_5\) by multiplying the fourth term \(a_4 = 0.216\) by the common ratio 0.3. Hence, \(a_5 = 0.216 \times 0.3 = 0.0648\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial component. It is the fixed number we multiply by to transition from one term to the next. Here, the common ratio is 0.3. This means each term is 30% of the previous one.
Understanding the common ratio helps us predict the direction and growth of the sequence:

• If the common ratio is greater than 1, the sequence grows exponentially.
• If it lies between 0 and 1, as in our example (0.3), the sequence shrinks.
• A negative common ratio can cause the sequence to alternate between positive and negative values.

Always double-check that you have the correct common ratio, as it deeply influences the entire sequence.

First Term

The first term, denoted as \(a_1\), is the starting point of any geometric sequence. It shapes the initial value from where the calculations proceed. In our exercise, the first term is 8.
Knowing the first term is vital for establishing a cohesive sequence of numbers. Here's why:

• It impacts all subsequent terms since each is derived from the one preceding it multiplied by the common ratio.
• Without the first term, it's impossible to calculate the rest of the sequence.

So, always start by clearly identifying your first term before moving into further calculations.

Term Calculation

Calculating terms in a geometric sequence involves multiplying the previous term by the common ratio. Let's see how it works:

• Second Term: Multiply the first term (8) by the common ratio (0.3) to get \( a_2 = 8 \times 0.3 = 2.4 \).
• Third Term: Continue by multiplying the second term (2.4) by the common ratio: \( a_3 = 2.4 \times 0.3 = 0.72 \).
• Fourth Term: Multiply the third term (0.72) by the common ratio: \( a_4 = 0.72 \times 0.3 = 0.216 \).
• Fifth Term: Finally, multiply the fourth term (0.216) by 0.3 to find \( a_5 = 0.216 \times 0.3 = 0.0648 \).

Repeat this process to find as many terms as needed by multiplying the latest term by the common ratio repeatedly.

Step by Step Solution

Breaking down the solution step by step makes it more accessible. Here’s how the sequence was calculated: - **Understanding the Sequence:** A geometric sequence needs the first term and a common ratio to find subsequent terms.- **Finding Each Term:**

• First Term (Given): Start with \( a_1 = 8 \).
• Second Term: Multiply by the common ratio: \( a_2 = a_1 \times 0.3 = 2.4 \).
• Third Term: \( a_3 = a_2 \times 0.3 = 0.72 \).
• Fourth Term: \( a_4 = a_3 \times 0.3 = 0.216 \).
• Fifth Term: \( a_5 = a_4 \times 0.3 = 0.0648 \).

Each step involves a straightforward multiplication, making this method ideal for arithmetic consistency.