Problem 31 - Algebra 2

Question

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. $$ 30,35,40,45, \dots $$

Step-by-Step Solution

Answer

The given sequence is arithmetic, and the next two terms are 50 and 55.

Step 1: Identify the Sequence Type

Let's determine if the sequence is arithmetic, geometric, or neither. For an arithmetic sequence, the difference between consecutive terms should be constant. In this case, observe if the difference between 35 and 30, and 40 and 35, and so on is the same. If so, it's an arithmetic sequence.

Step 2: Verify the Arithmetic Sequence

Subtract the second term from the first. If the difference is consistent in all terms, it's an arithmetic sequence.\n\nFirst Difference: \(35 - 30 = 5\)\nSecond Difference: \(40 - 35 = 5\)\nThird Difference: \(45 - 40 = 5\)\n\nThe difference is constant, so this is an arithmetic sequence.

Step 3: Find the Next Two Terms

To find the next terms in an arithmetic sequence, add the common difference to the last given term. The common difference here is 5. So, the next term after 45 would be \(45 + 5 = 50\), and the term after that would be \(50 + 5 = 55\).

Key Concepts

Sequences

Sequences are a collection of numbers ordered in a specific manner. Each number in the sequence is called a term. These terms follow certain rules or patterns, which help us predict future terms. Understanding these patterns is key to working with sequences effectively. In this exercise, we explored an ordered list of numbers: 30, 35, 40, 45, and so on. To identify the type of sequence, one must determine the relationship between the terms.

• In arithmetic sequences, the difference between consecutive terms is constant.
• Geometric sequences, on the other hand, have a constant ratio between terms.
• Some sequences might not fit either of these types, and are then classified as neither.

By identifying the sequence type, solving such problems becomes much easier.

Common Difference

The common difference is the heart of an arithmetic sequence. It refers to the constant amount that you add or subtract from one term to reach the next. In our example, the sequence was 30, 35, 40, 45. Let's discover the common difference here.

• Calculate the difference between successive terms, such as 35 - 30 = 5.
• Check this difference for the following pairs: 40 - 35 = 5 and 45 - 40 = 5.
• If the difference is constant, as it is here, it confirms the sequence is arithmetic.

The consistent difference of 5 in this sequence indicates it’s an arithmetic sequence. This allows us to predict the future terms with ease, knowing what value consistently changes the terms.

Term Identification

Term identification is crucial when working with sequences, as it involves recognizing specific terms and their positions in the sequence. Each number is a term, and commonly, these terms appear as a series of characters like a subscript or with the letter 'a.' For instance, in our series, 30 is the first term or a\( _1 \), 35 is a\( _2 \), and so forth. Identifying these terms makes it straightforward to apply sequence operations, such as finding the next terms or working backwards to find previous ones.

• Identify the position of each term in the sequence: first, second, third, etc.
• This helps in understanding the arithmetic operation needed for predicting subsequent or prior terms.
• In arithmetic sequences, the next term is always known by adding the common difference to the last known term.

Thus, term identification keeps your work organized and aids in solving more complex problems efficiently.

Problem Solving

Problem solving in sequences involves making use of the sequence type, common difference, and term identification to find unknown terms. Once you determine the pattern, you can predict subsequent terms or perhaps even missing terms in a long sequence. Using the original sequence: 30, 35, 40, 45, we've established the common difference as 5. To find the next two terms:

• Add the common difference to the last term, so 45 + 5 = 50.
• To find the next term after that, add the common difference to 50, making it 50 + 5 = 55.
• This logic can be extended to find as many terms as needed in an arithmetic sequence.

By applying these problem solving steps systematically, even complex sequences become manageable. It’s like following a recipe where each step leads you closer to the desired output.