Problem 67 - Prealgebra

Question

Find the next number in each sequence. $$2.5,2.75,3, \dots$$

Step-by-Step Solution

Answer

The next number in the sequence is 3.25.

Step 1: Identify the Pattern

Observe the given sequence: 2.5, 2.75, 3. Notice that each number is increasing. Calculate the difference between consecutive terms to find the pattern. \(2.75 - 2.5 = 0.25\) and \(3 - 2.75 = 0.25\). This indicates that each term increases by 0.25.

Step 2: Apply the Pattern

Once the pattern is identified as an increment of 0.25, apply this to the last known term in the sequence, which is 3. Add 0.25 to 3: \(3 + 0.25 = 3.25\).

Step 3: Verify Consistency

Check if applying the pattern consistently gives a logical sequence. So far, adding 0.25 after each term has worked. The following sequence: 2.5, 2.75, 3, 3.25 maintains the same increment pattern, confirming consistency.

Key Concepts

Pattern Identification

When facing an arithmetic sequence, spotting the pattern is the key first step. In this sequence, you start with the numbers 2.5, 2.75, and 3.

Recognizing how each number relates to the next helps in determining the rule they follow. Let's break this down:

• Find the difference between the first numbers, i.e., 2.75 minus 2.5. It results in 0.25.
• Verify this with the next step by subtracting 2.75 from 3, which also gives 0.25.

The pattern here is simple: each term increases by a constant value of 0.25. This pattern is crucial for predicting subsequent numbers in the sequence.

Incremental Steps

Once the pattern has been identified, the next phase is applying incremental steps to extend the sequence. Every number in an arithmetic sequence follows from adding a consistent amount, which we earlier identified as 0.25.

Here's how you apply it:

• Take the last known number in the series, which is 3.
• Add the incremental value we've identified, 0.25, to this number, resulting in 3.25.

By repeating this step, you can continue the sequence as needed. This process makes figuring out the sequence straightforward, no matter how many terms you're dealing with.

Sequence Consistency

After finding and applying the pattern, ensuring the sequence's consistency is vital. Why? Because consistency guarantees that the rule for the sequence holds across every term. To check consistency:

• Go through the sequence with the calculated increment (0.25 in this case) to ensure each step confirms the identified pattern.
• Look at the sequence: 2.5, 2.75, 3, 3.25. Verify each step involved the addition of 0.25 to continue the sequence accurately.

A consistent sequence is not only predictable but also aligns with the rule established from the pattern. This step ensures that no numbers deviate, which is crucial for error-free sequence progression.