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In the realm of geometric sequences, the concept of a common ratio is fundamental. It is the factor by which each term in the sequence is multiplied to achieve the next term. Understanding this concept makes it easier to comprehend the nature of geometric sequences. For the sequence at hand, the common ratio, denoted as \( r \), is given as \( \frac{1}{2} \). This means that each term in the sequence is half of the previous term.

• It scales the sequence
• It contributes to the exponential growth or decay
• It remains constant throughout the sequence

Recognizing the common ratio is crucial for predicting terms without re-calculating everything from scratch.

Sequence terms are the numbers that make up the geometric sequence. In our case, we are tasked to find the first five terms of the sequence where the first term \( a_1 \) is 20. The terms continue to be calculated by repeatedly applying the common ratio.

• The first term \( a_1 = 20 \)
• The second term \( a_2 = 20 \times \frac{1}{2} = 10 \)
• The third term \( a_3 = 10 \times \frac{1}{2} = 5 \)
• The fourth term \( a_4 = 5 \times \frac{1}{2} = 2.5 \)
• The fifth term \( a_5 = 2.5 \times \frac{1}{2} = 1.25 \)

These terms are derived from mathematical calculations guided by the initial term and the common ratio, highlighting how the sequence scales down progressively with each step.

Calculations involved in finding the terms of a geometric sequence are fairly straightforward but require careful attention to the operations. They mainly involve multiplication using the common ratio and each respective prior sequence term.
- **Multiplication with the common ratio:** By multiplying the prior term with the common ratio, the next term is determined. - **Consistent method:** For the first five terms, the arithmetic progression is carried out as follows: 1. Calculate the second term using \( a_1 \), multiply by \( r \) 2. For the third term using \( a_2 \) 3. Continue this pattern for subsequent terms
Effectively performing these calculations ensures accurate results, demonstrating how arithmetic operations are applied in a systematic manner to navigate through the sequence. Paying attention to each multiplication step solidifies understanding of the sequence dynamics.