Problem 3 - College Algebra

Question

Write the first five terms of each geometric sequence. $$ a_{1}=20, \quad r=\frac{1}{2} $$

Step-by-Step Solution

Answer

The first five terms of the sequence are: 20, 10, 5, 2.5, and 1.25.

Step 1: Determine the First Term

Since we are given the first term in the sequence, we can say that the first term, a_{1}, is 20.

Step 2: Use the Formula to Find the Second Term

To find the second term, plug in the values into the formula:\( a_{2} = a_{1} * r^{(2-1)} = 20 * (1/2)^{(2-1)} = 20 * (1/2) = 10 \).

Step 3: Find the Third Term

Again, using the formula but with n=3: \( a_{3} = a_{1} * r^{(3-1)} = 20 * (1/2)^{(3-1)} = 20 * (1/4) = 5 \).

Step 4: Find the Fourth Term

Following the same pattern, find the fourth term with n=4: \( a_{4} = a_{1} * r^{(4-1)} = 20 * (1/2)^{(4-1)} = 2.5 \).

Step 5: Find the Fifth Term

Finally, find the fifth term: \( a_{5} = a_{1} * r^{(5-1)} = 20 * (1/2)^{(5-1)} = 1.25 \).

Key Concepts

Understanding the First Term in a Geometric Sequence

In a geometric sequence, the first term plays a crucial role as it serves as the starting point for the entire sequence. It is often denoted as \(a_1\). This term is given at the beginning of your problem, making it your reference point for finding all subsequent terms in the sequence.
For instance, in our example, the first term \(a_1\) is 20. This means we will use the number 20 as a foundation to find the upcoming terms.
Knowing the first term allows you to understand the starting value without which the sequence cannot be determined. Essentially, it sets the stage for how the sequence will unfold.

Deciphering the Common Ratio in a Geometric Sequence

In geometric sequences, the common ratio \(r\) is the factor you multiply by to progress from one term to the next. The common ratio reveals the relationship between consecutive terms, highlighting how the sequence grows or shrinks.
Finding this ratio is crucial as it determines the nature of your geometric sequence:

• If \(r > 1\): the sequence grows exponentially, growing larger with each term.
• If \(0 < r < 1\): the sequence shrinks, with each term getting smaller.
• If \(r = 1\): each term is the same, resulting in a constant sequence.
• If \(r = 0\): every term after the first is zero.

In the exercise, our common ratio is \(\frac{1}{2}\). This means each term is half of the previous one, causing the sequence to diminish from the starting point of 20.

Exploring the Terms of a Geometric Sequence

The terms of a geometric sequence are determined using a specific formula that incorporates both the first term and the common ratio. The formula for the \(n\)-th term is:\[ a_n = a_1 \cdot r^{(n-1)} \]In this formula, \(a_n\) represents the term you want to find, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) denotes the position of the term within the sequence.
By substituting the relevant numbers into this formula, you can swiftly calculate any required term. For example:

• For the second term (\(a_2\)) in our exercise, we used: \(20 \cdot (\frac{1}{2})^{1} = 10\)
• For the third term (\(a_3\)): \(20 \cdot (\frac{1}{2})^{2} = 5\)

This methodology applies to find further terms, consistently multiplying by the common ratio raised to the relevant power.