Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, write the first five terms of the geometric sequence. \(a_{n}=12 \cdot\left(-\frac{1}{2}\right)^{n-1}\)

Step-by-Step Solution

Answer

The first five terms of the sequence are 12, -6, 3, -\frac{3}{2}, and \frac{3}{4}.

Step 1: Identify the first term of the sequence

The first term of the geometric sequence is found by substituting \(n = 1\) into the formula. Calculate \(a_1 = 12 \cdot \left(-\frac{1}{2}\right)^{1-1}\).

Step 2: Compute the first term \(a_1\)

Plug \(n = 1\) into the formula: \(a_1 = 12 \cdot \left(-\frac{1}{2}\right)^{0} = 12 \cdot 1 = 12\). The first term is \(12\).

Step 3: Compute the second term \(a_2\)

Substitute \(n = 2\) into the formula: \(a_2 = 12 \cdot \left(-\frac{1}{2}\right)^{2-1} = 12 \cdot \left(-\frac{1}{2}\right) = -6\).

Step 4: Calculate the third term \(a_3\)

For \(n = 3\), substitute into the formula: \(a_3 = 12 \cdot \left(-\frac{1}{2}\right)^{3-1} = 12 \cdot \left(\frac{1}{4}\right) = 3\).

Step 5: Determine the fourth term \(a_4\)

With \(n = 4\), the formula becomes: \(a_4 = 12 \cdot \left(-\frac{1}{2}\right)^{4-1} = 12 \cdot \left(-\frac{1}{8}\right) = -\frac{3}{2}\).

Step 6: Find the fifth term \(a_5\)

Substitute \(n = 5\) into the formula: \(a_5 = 12 \cdot \left(-\frac{1}{2}\right)^{5-1} = 12 \cdot \frac{1}{16} = \frac{3}{4}\).

Key Concepts

Sequence Formulas

A sequence formula is a mathematical expression that defines the nth term of a sequence. In our example, the formula used is:
\[ a_n = 12 \cdot \left(-\frac{1}{2}\right)^{n-1} \]
This formula helps determine any term in the sequence without having to list them all.

• \(a_n\) represents the nth term of the sequence.
• 12 is the initial term multiplier, which scales the result of the expression \((-\frac{1}{2})^{n-1}\).
• The exponent \(n-1\) signifies how many times the common ratio should be applied starting from the first term.

Understanding these components allows you to find any term within a geometric sequence easily. Integration of these formulas involves arithmetic operations like exponentiation, which must be handled carefully to ensure the correct calculations.

Terms of a Sequence

In sequences, each number is a term, and terms are arranged in a specific order based on their position. Here, "term" signifies a specific place in a sequence indicated by \(n\):

• The first term, \(a_1\), is the initial point, calculated as \(12 \times 1 = 12\).
• The second term, \(a_2\), calculated as \(12 \times (-\frac{1}{2}) = -6\), follows the progression.
• The third term, \(a_3\), becomes \(3\).
• The fourth term, \(a_4\), is \(-\frac{3}{2}\).
• Finally, the fifth term, \(a_5\), is \(\frac{3}{4}\).

Each of these terms is derived using the sequence formula by changing the value of \(n\). By understanding the concept of "terms," students can identify the position and value of elements within a sequence.

Geometric Progression

A geometric progression, also known as a geometric sequence, is a pattern of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the "common ratio."
In this sequence, the common ratio is \(-\frac{1}{2}\). This ratio affects how the sequence progresses:

• Starting from the first term \(12\), multiply it by \(-\frac{1}{2}\) to find the next term \(-6\).
• Continue multiplying each resulting term by \(-\frac{1}{2}\) to get the following terms: \(3\), \(-\frac{3}{2}\), and \(\frac{3}{4}\).

This progression alternates signs due to the negative ratio, creating a pattern of positive and negative terms. Understanding geometric progressions is crucial for solving problems related to exponential growth or decay, as it involves applying a common factor or rate repeatedly.