Problem 93 - Prealgebra

Question

Arithmetic Sequences Recall that an arithmetic sequence is a sequence in which each term comes from the previous term by adding the same number each time. For example, the sequence \(1, \frac{3}{2}, 2, \frac{5}{2}, \ldots\) is an arithmetic sequence that starts with the number 1. Then each term after that is found by adding \(\frac{1}{2}\) to the previous term. By observing this fact, we know that the next term in the sequence will be \(\frac{5}{2}+\frac{1}{2}=\frac{6}{2}=3\) Find the next number in each arithmetic sequence below. $$1, \frac{4}{3}, \frac{5}{3}, 2, \dots$$

Step-by-Step Solution

Answer

The next term is \( \frac{7}{3} \).

Step 1: Identify the Common Difference

In an arithmetic sequence, the common difference is the constant amount added to each term to get the next term. To find the common difference, subtract the first term from the second term: \( \frac{4}{3} - 1 = \frac{4}{3} - \frac{3}{3} = \frac{1}{3} \). Thus, the common difference between terms is \( \frac{1}{3} \).

Step 2: Calculate the Next Term

To find the next term in the sequence, add the common difference to the last known term in the sequence. The last term is \(2\), so add \( \frac{1}{3} \) to this term: \( 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \).

Step 3: Express the Next Term

Simplify the fraction to express the next term in the sequence in its simplest form. Already simplified as \( \frac{7}{3} \), this is the numerical representation of the next term in the sequence.

Key Concepts

Understanding Common Difference

In the world of arithmetic sequences, the term "common difference" refers to the constant amount that is added to each term to arrive at the subsequent one. Picture a staircase where each step is identical in height—this identical height represents the common difference in an arithmetic sequence.
For example, consider the sequence:

• First term: 1
• Second term: \(\frac{4}{3}\)

To find the common difference, subtract the first term from the second. So, \(\frac{4}{3} - 1 = \frac{1}{3}\). This \(\frac{1}{3}\) is the common difference and it tells us how much we need to add to each term to find the next one. Understanding this is crucial because it is the foundation of arithmetic sequences, allowing us to predict and calculate future terms.

Decoding Sequence Terms

The terms in a sequence are like the rungs on a ladder, and understanding each term is pivotal in navigating that ladder. In an arithmetic sequence, each term is derived by consistently adding the common difference to the previous term. Start from the beginning and keep moving forward.
Here's a snippet from our example sequence:

• First Term: 1
• Second Term: \(\frac{4}{3}\)
• Third Term: \(\frac{5}{3}\)

Notice how by adding the common difference of \(\frac{1}{3}\) to each current term, you land logically onto the next term. If you apply this consistently, you unravel the sequence and find the terms with ease. Keep in mind that while sequences may sound complex, they are essentially just patterned and sequential additions.

Mastering Fractions in Sequences

Fractions can often seem daunting, but in arithmetic sequences, they are simply another form of representing numbers. To integrate fractions into sequences, you need a little familiarity with fraction addition and simplification.
In our example, after identifying the common difference (\(\frac{1}{3}\)), you will apply this same difference even if the terms are fractions. As such:

• Adding fractions involves finding a common denominator, which simplifies the process significantly.
• A good rule of thumb is to imagine fractions as steps, where each addition is a small advancement along a number line.
• Even complicated arithmetic with fractions can be distilled to simple steps: add numerators, keep the common denominator, and simplify if necessary.

As showcased by the term \(2 + \frac{1}{3} = \frac{7}{3}\), mastering fractions is fundamentally about understanding these basics of fraction addition and ensuring accuracy by simplifying results when possible.