Problem 59 - Prealgebra

Question

Find the next term in each list. $$15,18,22,25,29, \dots$$

Step-by-Step Solution

Answer

The next term is 32.

Step 1: Identify the Pattern

Look at the differences between each consecutive term in the sequence. Calculate these differences to find a pattern.

Step 2: Calculate Differences

Subtract each term from the next term: - 18 - 15 = 3 - 22 - 18 = 4 - 25 - 22 = 3 - 29 - 25 = 4 The differences alternate between 3 and 4.

Step 3: Determine the Next Difference

Observe the alternating pattern: 3, 4, 3, 4, ..., so the next difference after 29 should be 3.

Step 4: Add the Next Difference to the Last Term

Add 3 to the last term, 29, to find the next term in the sequence: \[ 29 + 3 = 32 \]

Step 5: Verify the Result

Check that this new sequence: 15, 18, 22, 25, 29, 32, continues the pattern of alternating differences of 3 and 4.

Key Concepts

Pattern Recognition

Recognizing patterns is like solving a mystery. In arithmetic sequences, patterns help us predict the future terms. By looking at a sequence, the first step is to observe any recurring changes in the numbers.
For example, in a sequence like 15, 18, 22, 25, 29, keeping an eye on the differences between each term helps in spotting the pattern. Patterns may not always be straightforward, but they often follow a logical progression, such as consistent addition or subtraction between terms.
In our sequence, did you notice what was happening? There was a zigzag pattern alternating between adding 3 and adding 4. Such a pattern helps us set the stage for finding missing numbers.

Finding Differences

To analyze a sequence, calculate the difference between each term. It's a detective work done with simple subtraction. Here's how:

• Take two consecutive terms.
• Subtract the first number from the second one.

These calculations uncover the hidden steps taken to move from one term to another.
For the sequence 15, 18, 22, 25, 29:

• 18 - 15 = 3
• 22 - 18 = 4
• 25 - 22 = 3
• 29 - 25 = 4

Do you see how the sequence alternates? By finding differences, we verify if a sequence follows a particular pattern, assisting in further prediction of terms.

Sequence Continuation

Once you find the difference pattern, predicting further terms in a sequence becomes straightforward. It's like following stepping stones across a stream.
If a sequence such as 15, 18, 22, 25, 29 continues with differences of 3 and 4, you use the last known term and the next expected difference to determine the next number.
The last term here was 29, and the expected next difference was 3, so:\[ 29 + 3 = 32 \]Understanding this concept is crucial to not just complete puzzles, but also to apply mathematical ideas to real-life scenarios. Practicing with sequences hones your predictive skills.

Mathematical Reasoning

Mathematical reasoning is the backbone of solving sequences effectively. It means being logical and orderly in your approach
You use reasoning to deduce each step from pattern recognition to predicting terms.
In this exercise, the reasoning began by observing numbers, calculating differences, and once the pattern of alternating numbers was identified, applying logic to foresee future terms.
Using reasoning helps verify if the pattern holds true for the sequence. It ensures that your predictions maintain consistency:

• The sequence alternates between adding 3 and 4.
• So, after 29 with a difference of 3 gives us 32

Strong reasoning doesn't only apply to sequences; it applies to all areas of mathematics and problem-solving in life.