Problem 27 - College Algebra

Question

Write the first five terms of the sequence. $$a_{1}=3, a_{n}=(-3) a_{n-1}$$

Step-by-Step Solution

Answer

The first five terms are: 3, -9, 27, -81, and 243.

Step 1: Understand the First Term

The sequence starts with the first term given as \(a_1 = 3\). This is our initial value, and we will build upon this term to find subsequent terms.

Step 2: Find the Second Term

To find the second term \(a_2\), we use the relationship \(a_n = (-3) a_{n-1}\). Substituting \(n=2\) and \(a_1 = 3\), we get:\[a_2 = (-3) imes 3 = -9\]Thus, the second term is \(a_2 = -9\).

Step 3: Calculate the Third Term

Next, we find \(a_3\) by substituting \(a_2 = -9\) into the formula \(a_n = (-3) a_{n-1}\):\[a_3 = (-3) imes (-9) = 27\]Thus, the third term is \(a_3 = 27\).

Step 4: Determine the Fourth Term

Using the same formula, substitute \(a_3 = 27\) to find \(a_4\):\[a_4 = (-3) imes 27 = -81\]The fourth term is \(a_4 = -81\).

Step 5: Calculate the Fifth Term

Finally, substitute \(a_4 = -81\) to determine \(a_5\):\[a_5 = (-3) imes (-81) = 243\]Therefore, the fifth term is \(a_5 = 243\).

Key Concepts

Recursive Formula

A recursive formula is an important mathematical expression used to determine the terms of a sequence. Unlike explicit formulas, which express terms independently based on their position, recursive formulas rely on preceding terms to determine the next one. This relationship builds upon each term successively.
This particular formula, given for the geometric sequence, is: \[ a_n = (-3) \cdot a_{n-1} \]This means that to find any term \(a_n\), you multiply the previous term \(a_{n-1}\) by the constant -3. The constant is known as the multiplier or common ratio in the context of geometric sequences. This method of constructing a sequence emphasizes understanding the transformation between consecutive terms, which is crucial for identifying patterns.

Sequence Terms

In mathematics, sequences are ordered lists of numbers defined by an initial term and a rule or formula for finding subsequent terms. Understanding sequence terms, especially in a geometric sequence, is essential for mastering patterns and progression. - The first term, \(a_1\), is the foundation of the sequence. It is usually given or determined directly.- Using the recursive formula, each successive term is built from the previous one.- In the example sequence: - The first term is \(a_1 = 3\). - The second term, \(a_2 = -9\), is found by multiplying \(a_1\) by the negative multiplier (-3). - This process continues with \(a_3 = 27\), \(a_4 = -81\), and \(a_5 = 243\).
Knowing each term and how it's derived helps in understanding the broader properties of geometric sequences and their behaviors.

Negative Multiplier

A negative multiplier, in a geometric sequence, adds an interesting twist to the progression. This multiplier changes the sign of the terms as the sequence progresses.
In the given problem, the multiplier is -3. This results in the alteration of the sign of each subsequent term relative to the one before it. - If the previous term is positive, multiplying by -3 yields a negative next term. - Conversely, a negative term will become positive when multiplied by -3.
This alternating pattern between positive and negative terms highlights the dynamic nature of sequences with negative multipliers. Understanding this concept is key in solving various mathematical problems involving sequences where sign changes are intrinsic to their behavior.