Problem 2 - Beginning and Intermediate Algebra

Question

Fill in each blank with the correct response. In the sequence \(3,6,9,12,\) the term \(a_{3}=\) _____.

Step-by-Step Solution

Answer

9

Step 1: Identify the Given Sequence

The given sequence is 3, 6, 9, 12, ...

Step 2: Determine the Position of the Term

Identify the position of the term we need to find. Here, we need to find the 3rd term, which is represented as \( a_3 \).

Step 3: Read the Term at the Position

Counting the terms starting from the first, the 3rd term in the sequence is 9.

Key Concepts

Sequence Identification

To understand arithmetic sequences, let's start by identifying them. In an arithmetic sequence, each term after the first is obtained by adding a constant difference, called the 'common difference,' to the previous term.
For example, in the sequence 3, 6, 9, 12, the constant difference is 3. That's because:

• 6 - 3 = 3
• 9 - 6 = 3
• 12 - 9 = 3

This consistent difference tells us we have an arithmetic sequence. Knowing this helps us understand how the sequence progresses and allows us to predict future terms.

Term Position

Once we've identified the sequence type, the next step is to find the position of a specific term.
In this exercise, we're interested in the position of the 3rd term, which is represented as \(a_3\). Term positions are crucial for understanding and solving sequence problems because they tell us where a term stands in the sequence.
We count the terms from the first term (position 1) up to the desired term position. For our sequence 3, 6, 9, 12, we need the 3rd term. We count:

• 1st term: 3
• 2nd term: 6
• 3rd term: 9

Hence, the 3rd term here is 9.

Series Analysis

Series analysis delves deeper by examining how the terms relate as a whole. For arithmetic sequences, this involves recognizing patterns and the overall structure.
By understanding an arithmetic sequence's common difference and position, we can understand the whole sequence's behavior. For instance, knowing the common difference of 3 in our sequence 3,6,9,12,
helps us recognize that each term increases uniformly. This pattern is crucial for predicting future terms or identifying missing elements. Additionally, arithmetic sequences can be graphed linearly due to their consistent rate of change, making them easier to visualize and study.