Problem 20 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=10, a_{2}=30 $$

Step-by-Step Solution

Answer

The first five terms of the sequence are 10, 30, 90, 270, and 810.

Step 1: Identify three important elements of the geometric sequence

To find the first five terms of the geometric sequence, start by identifying the first term \(a_1\), the second term \(a_2\), and the common ratio \(r\). Here, \(a_1 = 10\) and \(a_2 = 30\).

Step 2: Calculate the common ratio \(r\)

The common ratio \(r\) of a geometric sequence is found using the formula \(r = \frac{a_2}{a_1}\). Substituting the given values, \(r = \frac{30}{10} = 3\).

Step 3: Use the common ratio to find subsequent terms

To find the next terms in the sequence, use the formula \(a_n = a_1 \cdot r^{(n-1)}\). Compute each term up to the fifth term.

Step 4: Calculate the sequence terms

Plug in the values:- The first term \(a_1 = 10\).- The second term \(a_2 = 30\).- The third term \(a_3 = 10 \cdot 3^2 = 10 \cdot 9 = 90\).- The fourth term \(a_4 = 10 \cdot 3^3 = 10 \cdot 27 = 270\).- The fifth term \(a_5 = 10 \cdot 3^4 = 10 \cdot 81 = 810\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key element that determines how the sequence progresses. It is the factor by which we multiply each term to get the next term in the sequence. To identify the common ratio, divide the second term by the first term.

For instance, if we have a sequence where the first term is 10 and the second term is 30, then the common ratio \( r \) is calculated as follows:
\[ r = \frac{30}{10} = 3 \]

This common ratio of 3 tells us that each term in this particular geometric sequence is obtained by multiplying the previous term by 3. Understanding the common ratio is essential for predicting the sequence's behavior and calculating future terms.

First Term of Sequence

The first term of a geometric sequence, often denoted as \( a_1 \), serves as the starting point of the entire sequence. It plays a crucial role because every subsequent term in the sequence is derived from it, using the common ratio.

In our given sequence, the first term is 10. This means that 10 is the number we begin with before applying the common ratio to find out the following terms of the sequence.

The value of the first term can significantly influence the sequence's growth or decay, particularly with larger common ratios. Therefore, grasping the significance of the first term is crucial to understanding the nature of the full sequence.

Formula for nth term

To find any term in a geometric sequence, we can use a powerful mathematical formula. This formula helps us calculate the \( n \)-th term without having to list out all the previous terms:
\[ a_n = a_1 \cdot r^{(n-1)} \]

Where:

• \( a_n \) is the \( n \)-th term we want to find
• \( a_1 \) is the first term of the sequence
• \( r \) is the common ratio
• \( n \) is the position of the term in the sequence

This formula is particularly useful because it lets us jump directly to any term in the sequence without performing repetitive calculations for each interim term. For example, if you want to find the fifth term in a sequence where \( a_1 = 10 \) and \( r = 3 \), you'd plug these into the formula:
\[ a_5 = 10 \cdot 3^{(5-1)} = 10 \cdot 81 = 810 \]

Sequence Calculation Steps

Calculating the terms of a geometric sequence involves several straightforward steps. Following these can help in understanding the progression of the sequence and in finding the required terms easily:

• Step 1: Identify key elements: Start by recognizing the first term \( a_1 \), and the second term \( a_2 \), which will give us the common ratio.
• Step 2: Determine the common ratio: Use the formula \( r = \frac{a_2}{a_1} \) to find the common ratio, which is the factor for multiplying each term to get to the next.
• Step 3: Apply the common ratio: Use the formula for the \( n \)-th term \( a_n = a_1 \cdot r^{(n-1)} \) to compute any term in the sequence.
• Step 4: Calculate the terms: Plug in the values into the formula to calculate each term step by step, up to the required number of terms.

In the example provided, the calculations yield the following terms for the sequence: \(10, 30, 90, 270, 810\). Each term is derived effortlessly using these steps, aiding both understanding and efficiency.