Problem 21 - Beginning and Intermediate Algebra

Question

Find the general term, \(a_{m}\) for each geometric sequence. Then, find the indicated term. $$a_{1}=-\frac{1}{2}, r=-\frac{3}{2} ; a_{4}$$

Step-by-Step Solution

Answer

The general term for the geometric sequence is \(a_{m} = -\frac{1}{2} \cdot \left(-\frac{3}{2}\right)^{m-1}\). Using this formula, the fourth term (\(a_4\)) is \(-\frac{27}{16}\).

Step 1: Identify Given Values

From the problem, we have the initial values: $$a_1 = -\frac{1}{2}$$ $$r = -\frac{3}{2}$$ We are tasked to find the fourth term (\(a_4\)) in the geometric sequence.

Step 2: Use the Formula to Find the General Term

Use the general formula for any term in a geometric sequence: $$a_{m} = a_{1} \cdot r^{m-1}$$ Plug in the given values for \(a_1\) and \(r\): $$a_{m} = -\frac{1}{2} \cdot \left(-\frac{3}{2}\right)^{m-1}$$ Now we have our general term formula in terms of \(m\).

Step 3: Find the Fourth Term

Use the formula found in step 2 to calculate the fourth term (\(a_4\)) by substituting \(m = 4\): $$a_{4} = -\frac{1}{2} \cdot \left(-\frac{3}{2}\right)^{4-1}$$ Simplify and calculate the result: $$a_{4} = -\frac{1}{2} \cdot \left(-\frac{3}{2}\right)^{3}$$ $$a_{4} = -\frac{1}{2}\cdot\left(-\frac{27}{8}\right)$$ $$a_{4} = -\frac{27}{16}$$ So, the fourth term (\(a_4\)) in the given geometric sequence is \(\boxed{-\frac{27}{16}}\).

Key Concepts

General Term

The general term of a geometric sequence is like a magic formula that helps you determine any term within the sequence. It is incredibly useful because once you have it, you can find any term without having to manually calculate each preceding term. The formula for the general term is given by: \( a_{m} = a_{1} \cdot r^{m-1} \), where:

• \( a_{m} \) represents the m-th term you want to find.
• \( a_{1} \) is the first term of the sequence.
• \( r \) is the common ratio, which is the factor by which each term is multiplied to get the next term.
• \( m \) is the term position you are interested in.

To use this formula effectively, you need to know the first term and the common ratio of the sequence. Once you have these, simply plug them into the formula along with the position of the term you want to find. This will give you any term in the sequence directly.

Fourth Term

Finding the specific term you're interested in, such as the fourth term in this case, is straightforward once you know how to use the general term formula. Suppose you have already derived the general term formula for the sequence: \( a_{m} = -\frac{1}{2} \cdot \left(-\frac{3}{2}\right)^{m-1} \). To find the fourth term, \( a_4 \), you simply substitute 4 for \( m \).Plug the value into the formula:

• Calculate: \( a_{4} = -\frac{1}{2} \cdot \left(-\frac{3}{2}\right)^{4-1} \)
• Simplify the exponent: \( a_{4} = -\frac{1}{2} \cdot \left(-\frac{3}{2}\right)^3 \)
• Calculate the power: \( \left(-\frac{3}{2}\right)^3 = -\frac{27}{8} \)
• Finish the calculation: \( a_{4} = -\frac{1}{2} \cdot -\frac{27}{8} = -\frac{27}{16} \)

Thus, the fourth term of the sequence is \( -\frac{27}{16} \). This shows the power of the general term formula, as you can bypass manually listing several terms.

Geometric Sequence Formula

The geometric sequence formula is the backbone of finding terms within geometric sequences. It provides a structured way to calculate any term by focusing on the relationship between terms rather than listing them. The essential part of this formula is the common ratio \( r \), which measures how each term increases or decreases in the sequence based on multiplication.Here are key points about the formula:

• The formula is \( a_{m} = a_{1} \cdot r^{m-1} \).
• It applies to any geometric sequence and helps find terms quickly.
• Knowing just two pieces of information—the first term \( a_{1} \) and the common ratio \( r \)—allows you to explore the entire sequence.

The efficiency and elegance of the formula make it an essential tool in algebra, enabling quick calculations of specific terms in sequences without tedious repetition.