Problem 44 - Prealgebra

Question

Find the next term in the pattern. $$326,344,362,380, \dots$$

Step-by-Step Solution

Answer

The next term is 398.

Step 1: Identify the Pattern

First, look at the differences between consecutive terms. Calculate the differences: 344 - 326 = 18, 362 - 344 = 18, 380 - 362 = 18. It seems the difference between each consecutive term is consistently 18.

Step 2: Use the Pattern to Find the Next Term

Since the difference between each term is 18, add 18 to the last known term in the sequence. The last known term is 380. 380 + 18 = 398.

Key Concepts

Patterns

Patterns in arithmetic sequences are like a set of instructions showing how to move from one number to the next in a sequence. In this case, each term in the sequence increases by a certain amount, a regular step if you will.
This regular increase, or pattern, helps us predict what comes next.

Identifying the pattern requires a bit of detective work. We look at the sequence of numbers and find the rule that is being applied to go from one number to another.
This means finding the common difference between them, which we can then use to solve for the next terms.

Once you recognize this common difference, the pattern becomes clear, and extending the sequence is straightforward.
These types of patterns, where each term is a specific distance away from its neighbor, are characteristic of arithmetic sequences.
Remember, identifying the pattern is the key to unlocking the rest of the sequence.

Consecutive Terms

Consecutive terms are numbers that follow one another in a sequence. In arithmetic sequences, consecutive terms have a consistent relationship defined by the pattern or common difference. Understanding this relationship is crucial for predicting future terms in the sequence.

In the given sequence, the terms are 326, 344, 362, and 380.
The relationship between each pair of consecutive terms informs us about this common difference. Recognizing that the difference is constant simplifies the task of finding any subsequent terms.
This constant difference gives structure and makes arithmetic sequences predictable.

By comprehending how each term relates to its predecessors, a clear picture of the sequence emerges, guiding us effortlessly to the next number.

Difference Calculation

Calculating the difference between consecutive terms is a fundamental step in identifying arithmetic sequences. This difference shows how much one term increases to get to the next.
It's essentially the key that unlocks the pattern of the sequence.

The difference is found by subtracting a term from the next consecutive term. For this sequence, observe the differences:

• 344 - 326 = 18,
• 362 - 344 = 18,
• 380 - 362 = 18.

These calculations show that the sequence increases by 18 each time.

This is called the "common difference". Once found, it not only confirms the pattern but also aids in calculating the next terms by simply adding it to the last known term.
The importance of recognizing and calculating this consistent difference cannot be overstated, as it turns the complexity of sequences into simple arithmetic.