Problem 6 - Algebra and Trigonometry

Question

Write the first five terms of each geometric sequence. $$ a_{n}=-3 a_{n-1}, \quad a_{1}=10 $$

Step-by-Step Solution

Answer

The first five terms of the given geometric sequence are 10, -30, 90, -270, 810.

Step 1: Identify the first term and the common ratio

In this geometric sequence, the first term \( a_{1} \) is given as 10 and the common ratio (which is the coefficient of \( a_{n-1} \) in the expression for \( a_{n} \)) is -3.

Step 2: Generate the second term

Use the given rule \( a_{n} = -3a_{n-1} \) to generate the second term. Substituting 1 for n-1 in the rule gives \( a_{2} = -3a_{1} = -3*10 = -30 \). Thus, the second term in the sequence is -30.

Step 3: Generate the third term

Again using the rule \( a_{n} = -3a_{n-1} \) to generate the third term. Substituting 2 for n-1 in the rule gives \( a_{3} = -3a_{2} = -3*(-30) = 90 \). Thus, the third term in the sequence is 90.

Step 4: Generate the fourth term

Again using the rule \( a_{n} = -3a_{n-1} \) to generate the fourth term. Substituting 3 for n-1 in the rule gives \( a_{4} = -3a_{3} = -3*90 = -270 \). Thus, the fourth term in the sequence is -270.

Step 5: Generate the fifth term

Finally, use the rule \( a_{n} = -3a_{n-1} \) to generate the fifth term. Substituting 4 for n-1 in the rule gives \( a_{5} = -3a_{4} = -3*(-270) = 810 \). Thus, the fifth term in the sequence is 810.

Key Concepts

Common Ratio

In a geometric sequence, one of the key terms you'll encounter is the "common ratio." It's like the magic number that helps you generate the sequence. Think of it as the multiplier you use each time to get from one term to the next.
For example, in the sequence described in the exercise, the common ratio is -3. This means that each term is obtained by multiplying the previous term by -3.

• If you have the first term as 10, then the second term will be 10 multiplied by -3, which is -30.
• The third term will be -30 multiplied by -3, making it 90.

The common ratio can be a positive or negative number, or even a fraction, and it determines how the sequence behaves as it progresses. When dealing with a negative common ratio, you'll notice the sequence alternating between positive and negative numbers.

First Term

When diving into a geometric sequence, the first term is your starting point. You typically represent it with the symbol \( a_1 \). The first term sets the stage for the rest of the sequence and is the number you'll initially work from.
In the example provided, the first term is given as 10. This means you begin your sequence with the number 10, which you will then multiply by the common ratio to find subsequent terms.
The value of the first term will directly impact all the terms following in the sequence. Understanding this number is crucial, as it's your "launchpad" for the entire progression.

Sequence Generation

Generating a sequence involves a process that takes you step-by-step through the pattern, using the starting point and the multiplier. This method is known as "sequence generation."
To generate a geometric sequence:

• Start with the first term, which can be any numerical value. For our example, it's 10.
• Use the common ratio. Multiply the first term by the common ratio to find the next term, which is -3 for this sequence.
• Continue applying the same operation to determine each subsequent term. Each term is obtained by multiplying the preceding term by the common ratio.

The result is a sequence that follows a predictable pattern, allowing us to easily find any term in the progression. This system of sequence generation uses predefined rules to explore further numbers down the line.

Geometric Progression

A geometric sequence is often referred to as a "geometric progression," which highlights its continuous and structured nature. It's a mathematical arrangement where each term after the first is produced by multiplying the previous one by a constant called the "common ratio."
A geometric progression, as reflected in the solved example, showcases a clear recurring pattern.

• The sequence starts with the number 10.
• Each subsequent term is the previous term multiplied by -3, consistently following this rule.

Understanding geometric progression helps in many fields, like finance for calculating interest, physics for modeling exponential growth or decay, and much more. It visually and numerically demonstrates how repeated multiplication affects number sequences.