Problem 4 - Algebra 2

Question

Is the sequence geometric? If so, find the common ratio and the next two terms. $$ -1,1,-1,1, \ldots $$

Step-by-Step Solution

Answer

Yes, the given sequence is geometric with a common ratio of -1. The two next terms of the sequence are -1 and 1.

Step 1: Identify the sequence

Observe the given sequence which is -1, 1, -1, 1... Find a pattern or common factor between consecutive terms.

Step 2: Validate if the sequence is geometric

Check if every term in the sequence is the result of multiplying the previous term with a specific constant. Here, every term is the product of the previous term and -1.

Step 3: Define the common ratio

Since every term is the product of the previous term and -1, the common ratio for this geometric sequence is -1.

Step 4: Compute the next two terms

The next terms can be found by multiplying the last known term by the common ratio. Here, the last term we know is 1; so, next two terms are \(1 \times -1 = -1\) and \(-1 \times -1 = 1\).

Key Concepts

Common Ratio

A geometric sequence is one where each term after the first is found by multiplying the previous term by a constant, known as the "common ratio."
For any sequence to be geometric, this ratio must remain consistent throughout.
In the example given, the sequence is

• -1,
• 1,
• -1,
• 1, ...

By observing the changes between these terms, we can identify the common ratio is

• When -1 is multiplied by -1, the result is 1
• Similarly, 1 multiplied by -1 returns -1 again.

Thus, the constant factor is

• -1,

which confirms the sequence as geometric.
This factor is crucial for forecasting all subsequent terms in the sequence.

Sequence Pattern

Understanding the pattern in a sequence helps to determine if it is geometric or not.
In a geometric sequence, one should look for a recurrent pattern that dictates that each term is a product of the previous term and a constant factor, the common ratio.
In the given sequence:

• The pattern alternates between -1 and 1.

This specific pattern of the sequence is due to the common ratio of -1.
Each term in the sequence affects the next in a consistent manner, following the multiplication by -1 from one term to the next.
This ensures that the pattern will indefinitely alternate.
Such a clear pattern is evidence that the terms are related linearly, affirming they belong to a geometric series.

Multiplying Terms

The multiplication of terms is the key process driving a geometric sequence.
Each new term is attained by multiplying the previous one by the common ratio.
For example, consider multiplying terms in the given sequence:

• Start with the term -1, the first multiplication by the common ratio -1 gives 1.
• Continuing, the next multiplication, 1 by -1, reverts the term back to -1.

This repetitive process of multiplying by -1 highlights how each term is interlinked through simple multiplication.
To extend the sequence:

• Begin with the last known term, 1, multiply by the common ratio -1 to get -1.
• Then, multiply -1 by -1 again to end at 1.

By repeating these steps, you can generate additional terms easily, maintaining the sequence's geometric integrity overall.