Problem 17 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=2, r=3 $$

Step-by-Step Solution

Answer

The first five terms are 2, 6, 18, 54, 162.

Step 1: Understanding the Problem

We need to find the first five terms of a geometric sequence. We are given the first term, \( a_1 = 2 \), and the common ratio, \( r = 3 \).

Step 2: Formula for the nth Term

In a geometric sequence, the \( n \)-th term is given by the formula: \( a_n = a_1 \, r^{n-1} \). We will use this formula to calculate each of the first five terms.

Step 3: Calculate the First Term

The first term is already given: \( a_1 = 2 \).

Step 4: Calculate the Second Term

Using the formula \( a_n = a_1 \, r^{n-1} \), calculate \( a_2 \): \( a_2 = 2 \, (3)^1 = 2 \, \times \, 3 = 6 \).

Step 5: Calculate the Third Term

Calculate \( a_3 \) using the formula: \( a_3 = 2 \, (3)^2 = 2 \, \times \, 9 = 18 \).

Step 6: Calculate the Fourth Term

Calculate \( a_4 \): \( a_4 = 2 \, (3)^3 = 2 \, \times \, 27 = 54 \).

Step 7: Calculate the Fifth Term

Calculate \( a_5 \): \( a_5 = 2 \, (3)^4 = 2 \, \times \, 81 = 162 \).

Step 8: List the First Five Terms

The first five terms of the sequence are: 2, 6, 18, 54, 162.

Key Concepts

Common Ratio

In a geometric sequence, understanding the common ratio is crucial. It tells you how each term relates to the one before it. You get it by dividing any term by the term that comes before it. For instance, with the sequence starting at 2 and having terms 2, 6, 18, and so on, the common ratio is 3. So, each term is three times the one before it.

• If you have the sequence 2, 6, 18, 54, the common ratio, 3, means: 6 is 3 times 2, 18 is 3 times 6, and so forth.
• The common ratio keeps the sequence growing or shrinking at a consistent rate.

This consistency is what defines the pattern of the geometric sequence. If you ever get lost, just redivide one term by the previous one to find your way back to the common ratio.

Nth Term Formula

The nth term formula in a geometric sequence helps determine the specific position of any term without listing all previous ones. The formula is:\[ a_n = a_1 \, r^{n-1} \]Where:

• \( a_n \) is the term you want to find.
• \( a_1 \) is the first term in the sequence.
• \( r \) is the common ratio.
• \( n \) is the term number.

For instance, if you want the third term of the sequence, you only need the first term (2) and the ratio (3):\[ a_3 = 2 \, (3)^{2} = 18 \] Using this formula means you can find any term in just a few calculations, saving time.

Sequence Calculation

To calculate terms in a geometric sequence, you repeatedly use the nth term formula. This method ensures accuracy because each calculation is based on solid mathematics. Ensuring your calculations are correct means understanding the base term and the common ratio well. Here’s the step-by-step to find terms:

• Start with the given first term, \( a_1 \). For example, 2.
• Use the common ratio to find subsequent terms by applying \( a_n = a_1 \, r^{n-1} \).
• Adjust the exponent in \( r^{n-1} \) according to the term number you find.

Let's calculate terms again: 1. First term: \( a_1 = 2 \)2. Second term: \( a_2 = 2 \, (3)^1 = 6 \)3. Third term: \( a_3 = 2 \, (3)^2 = 18 \)4. Fourth term: \( a_4 = 2 \, (3)^3 = 54 \)5. Fifth term: \( a_5 = 2 \, (3)^4 = 162 \)Each step involves multiplication using powers of the common ratio, which reveals the beauty and multiplicative nature of geometric sequences.

First Five Terms

Now that we know how to calculate the terms, let's focus on the first five terms of a geometric sequence. We already have the first term, which is 2. Starting from it, we apply the common ratio iteratively using the nth term formula.

• First term, \( a_1 \), is given as 2.
• Second term, \( a_2 = 2 \, (3)^{1} = 6 \).
• Third term, \( a_3 = 2 \, (3)^{2} = 18 \).
• Fourth term, \( a_4 = 2 \, (3)^{3} = 54 \).
• Fifth term, \( a_5 = 2 \, (3)^{4} = 162 \).

These steps emphasize how each term builds off the last, thanks to the constant rate of growth specified by the common ratio. By focusing on these early terms, you gain insight into how the sequence unfolds and grows dramatically.