Problem 41 - Algebra 2

Question

Find the next two terms of each geometric sequence. $$ 1.4,-3.5,8.75, \dots $$

Step-by-Step Solution

Answer

The next two terms are \(-21.875\) and \(54.6875\).

Step 1: Identify the Common Ratio

To find the common ratio in a geometric sequence, divide the second term by the first term. Here, the second term is \(-3.5\) and the first term is \(1.4\). So, the common ratio \(r\) is \(-3.5/1.4 = -2.5\).

Step 2: Verify the Common Ratio

To ensure the correct common ratio, confirm that the third term is also obtained by multiplying the second term by the common ratio. Multiply \(-3.5\) by \(-2.5\) to get \(8.75\). Both calculations confirm the common ratio is \(-2.5\).

Step 3: Calculate the Fourth Term

The fourth term of the sequence is obtained by multiplying the third term \(8.75\) by the common ratio \(-2.5\). This results in \(-21.875\).

Step 4: Calculate the Fifth Term

The fifth term of the sequence is obtained by multiplying the fourth term \(-21.875\) by the common ratio \(-2.5\). This results in \(54.6875\).

Key Concepts

Common Ratio

A geometric sequence can be identified and defined by its common ratio. This ratio represents how each term in the sequence relates to the last. It is the key factor that allows us to predict any subsequent terms based on any preceding ones. To find this common ratio, you simply divide one term by the term preceding it. For example, take the sequence 1.4, -3.5, 8.75. To identify its common ratio, divide -3.5 (the second term) by 1.4 (the first term), resulting in a ratio of -2.5.
When working with geometric sequences, the common ratio can be very telling.

• If the common ratio is positive and less than one, terms steadily decrease.
• If greater than one, terms increase.
• Negative ratios lead to alternating positive and negative terms.

Marking the importance of this factor helps to correctly predict the flow or behavior of the sequence.

Terms Calculation

Once the common ratio is known, you can calculate further terms in the sequence easily by multiplying the prior term by this common ratio. For example, consider the above sequence with a common ratio of -2.5. To find the next term after 8.75:

• Multiply 8.75 by -2.5, yielding the next term, -21.875.
• Repeat this process, multiplying -21.875 by -2.5 to obtain 54.6875.

These calculations show the elegance of geometric sequences—each term builds upon the last in a predictable fashion. This process is straightforward yet effective for unveiling the forthcoming terms in the sequence.

Algebraic Sequence Analysis

Analyzing a sequence using algebra involves understanding its underlying pattern or rule. In a geometric sequence, this analysis centers around recognizing the formula for the nth term. The general formula for a geometric sequence is given by \( a_n = a_1 \times r^{(n-1)} \), where:

• \( a_n \) is the nth term,
• \( a_1 \) is the first term,
• \( r \) is the common ratio, and
• \( n \) is the term number.

Applying this knowledge to the sequence 1.4, -3.5, 8.75, the rule allows us to predict any term. Suppose you want the sixth term; substitute into the formula: \( 1.4 \times (-2.5)^5 \). This method highlights how algebraic analysis affords efficient and accurate extension beyond the numbers initially given, proving invaluable in various mathematical applications.