Problem 57 - Prealgebra

Question

Find the next term in each list. $$20,33,46,59,72, \dots$$

Step-by-Step Solution

Answer

The next term is 85.

Step 1: Identify the Pattern

First, examine the sequence to identify a pattern or rule that generates the next term. The first few terms are 20, 33, 46, 59, 72. We notice that each term is formed by adding a constant difference to the previous term.

Step 2: Calculate the Common Difference

Calculate the difference between consecutive terms to see if the same value is added each time. Compute: \( 33 - 20 = 13 \), \( 46 - 33 = 13 \), \( 59 - 46 = 13 \), \( 72 - 59 = 13 \). The common difference is 13, confirming it's an arithmetic sequence.

Step 3: Apply the Arithmetic Sequence Formula

To find the next term, use the formula for the nth term of an arithmetic sequence: \( a_n = a_1 + (n-1) imes d \). Here, the last known term (72) is at the 5th position, with \( d = 13 \). To find the 6th term, calculate \( a_6 = 72 + 13 = 85 \).

Step 4: Verify the Solution

Recheck the calculation to ensure no errors occurred:Add the common difference to the last given term: \( 72 + 13 = 85 \).This confirms that 85 is the correct next term in the sequence.

Key Concepts

Understanding the Common Difference

In an arithmetic sequence, a key element to grasp is the **common difference**. It’s the fixed amount added to each term to get the next term in the sequence. Understanding it is crucial because it dictates how the sequence progresses.
To identify the common difference, look at any two consecutive terms in the sequence and subtract the first from the second. For example, with numbers like 20, 33, 46, and so on, compute:

• 33 - 20 = 13
• 46 - 33 = 13
• 59 - 46 = 13

These calculations show that the common difference is 13. Once found, this consistent number helps in predicting future terms. Always confirm this consistency across multiple steps to ensure accuracy.

Recognizing Sequence Patterns

Sequences often follow distinct patterns. Specifically for arithmetic sequences, recognizing these patterns involves understanding their linear progression. This means each term is simply the previous term increased by a constant value—the common difference.
Observing the progression in our example sequence of 20, 33, 46, 59, and 72, we see that each number comes from adding 13, showing clear linear movement.
Patterns are foundational because they allow us to foresee how the sequence will evolve, making sequences predictable.

• Notice the repetitive step (adding 13, always the common difference)
• This repetition results in a steady increase in the sequence’s terms

Identifying these patterns early makes it easier to spot errors and confirms that the solution path is correct.

Finding the nth Term in an Arithmetic Sequence

The formula for finding the nth term in an arithmetic sequence provides a direct way to predict any term's value using algebra. The formula is:\[ a_n = a_1 + (n-1) \times d \] where:

• \( a_n \) is the nth term
• \( a_1 \) is the first term
• \( n \) is the term number
• \( d \) is the common difference

For the sequence 20, 33, 46, 59, 72 with common difference 13, suppose you need the 6th term. The first term, \( a_1 \), is 20 and \( d \) is 13. Plugging into the formula:\[ a_6 = 20 + (6-1) \times 13 \] which yields:\[ a_6 = 20 + 65 = 85 \] thus confirming the calculation. Mastering this formula means you can find any term in the sequence without listing each preceding term, saving time and reducing errors.