Problem 1 - Algebra and Trigonometry

Question

Write the first five terms of each geometric sequence. $$ a_{1}=5, \quad r=3 $$

Step-by-Step Solution

Answer

The first five terms of the geometric sequence are 5, 15, 45, 135, 405

Step 1: Identify the given values

We are given the first term of the sequence (\(a_{1}\)) is 5 and the common ratio (\(r\)) is 3.

Step 2: Apply the formula to find the second term

We use the formula \(a_2 = a_{1} \cdot r^{2-1} = 5 \cdot 3^{1} = 15\). So, the second term in the sequence is 15.

Step 3: Apply the formula to find the third term

Again, apply the formula \(a_3 = a_{1} \cdot r^{3-1} = 5 \cdot 3^{2} = 45\). The third term in the sequence is 45.

Step 4: Apply the formula to find the fourth term

Use the formula \(a_4 = a_{1} \cdot r^{4-1} = 5 \cdot 3^{3} = 135\). The fourth term in the sequence is 135.

Step 5: Apply the formula to find the fifth term

With the help of the formula \(a_5 = a_{1} \cdot r^{5-1} = 5 \cdot 3^{4} = 405\), the fifth term in the sequence is 405.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial factor. It represents the constant factor by which we multiply each term to get the next one. This ratio remains the same throughout the sequence.
An easy way to identify the common ratio is by dividing any term in the sequence by the previous term. For example, if the first term is 5 and the common ratio is 3, multiplying 5 by 3 gives us the next term which is 15.

• It's a vital part of understanding how geometric sequences grow or shrink.
• A large common ratio can lead to a sequence that grows quickly, while a small ratio (less than 1 but greater than 0) makes the sequence decrease.
• If the common ratio is negative, the sequence will alternate signs.

Understanding the common ratio aids in predicting future terms after the initial few terms are established.

Sequence Terms

The sequence terms are individual numbers that make up a geometric sequence. Each of these terms is calculated using the first term and the common ratio.
These terms are typically denoted as \(a_1, a_2, a_3, \ldots\). In our example, we begin with \(a_1 = 5\).
By applying the common ratio consistently, we can generate subsequent terms. This allows a sequence to follow a clear pattern:

• The second term, \(a_2 = 15\), follows \(a_1\).
• The third term, \(a_3 = 45\), follows \(a_2\), and so on.

Mastery of these terms helps in identifying geometric sequences and predicting their behavior across different contexts.

Mathematical Formula

To find terms in a geometric sequence, we use a mathematical formula. The standard formula for the \(n\)-th term in geometric sequences is \(a_n = a_1 \, r^{n-1}\). This allows us to compute any term without needing to calculate all previous terms.
Let's break down this formula:

• \(a_1\) is the first term, which sets your sequence's starting point.
• \(r\) represents the common ratio, providing the factor for each subsequent term.
• \(n\) is the position of the term in the sequence (such as the 2nd, 3rd, etc.)

For example, to find the second term using the formula, we substitute \(a_1 = 5\) and \(r = 3\): \(a_2 = 5 \times 3^{2-1} = 15\). This formula is a powerful tool for quickly accessing any term without manual multiplication at each step.

Term Calculation

The process of term calculation involves using the formula to determine individual terms within a geometric sequence. This method is efficient and prevents errors as manual calculations for larger sequences can become cumbersome.
Here's how you can apply it to calculate terms step by step:

• Ensure the first term \(a_1\) and common ratio \(r\) are known.
• Apply the formula \(a_n = a_1 \, r^{n-1}\) for each desired term.
• For our sequence, \(a_2 = 5 \times 3^{1} = 15\), \(a_3 = 5 \times 3^{2} = 45\).

Continue this pattern for each term you want to calculate.

By mastering term calculation within geometric sequences, you'll confidently handle more advanced problems. Embracing this concept enhances mathematical problem-solving skills.