Problem 13 - Prealgebra

Question

Describe each sequence using words and symbols. $$15,16,17,18, \dots$$

Step-by-Step Solution

Answer

The sequence starts at 15 and increases by 1 each time.

Step 1: Identify the Pattern

The given sequence is 15, 16, 17, 18, and so on. Notice how each number increases by 1 from the previous number, indicating a consistent pattern.

Step 2: Express the Sequence Using Words

In words, we can describe the sequence as 'starting at 15 and increasing by 1 each time.' This implies a linear and regular pattern in the sequence.

Step 3: Express the Sequence Using Symbols

We can describe the sequence mathematically using the general form of an arithmetic sequence: \(a, a+d, a+2d, a+3d, \ldots\), where \(a\) is the first term and \(d\) is the common difference. Here, \(a = 15\) and \(d = 1\). Thus, the sequence can be expressed as: \(15, 15+1, 15+2, 15+3, \ldots\).

Key Concepts

Pattern Recognition

Recognizing patterns in sequences is like finding order in a series of numbers. This is a useful skill because it helps solve problems and predict future terms. In an arithmetic sequence, numbers follow a pattern by adding the same amount repeatedly.

In our example, the numbers are 15, 16, 17, 18, and so on. Notice each number increases by 1 compared to the previous one. This consistent increment tells us it's an arithmetic sequence.

To spot such a pattern, observe if there's a regular interval or step from one number to the next. Here’s how you can do it:

• Look at each pair of numbers in the sequence.
• Check if the difference between consecutive numbers is constant.

Once you detect this regular step, you've identified the arithmetic pattern.

Sequence Description

Describing a sequence in words means explaining the order and arrangement of numbers. In our exercise, we start with the number 15. Then, we add 1 to each subsequent number.

This can be simplified and expressed in words as **'starting at 15 and increasing by 1 each time.'** Such descriptions help to quickly understand how the sequence behaves.

By clearly laying out the pattern using simple language, anyone can grasp the essence of the sequence without needing complex math terms. Remember, an arithmetic sequence is always about regular increases or decreases, making them easy to follow and describe.

Mathematical Notation

Mathematical notation allows us to express sequences concisely using symbols. For arithmetic sequences, use the format: \(a, a+d, a+2d, a+3d, \ldots\) where:

• \(a\) is the first term.
• \(d\) is the common difference.

In our example, 15 is the starting point (\(a\)), and we add 1 each time (\(d\)).

So, in mathematical terms, this sequence is expressed as: \(15, 15+1, 15+2, 15+3, \ldots\). This notation is efficient, showing how each term is related to its predecessor.

Writing sequences with symbols clarifies their structure, making calculations straightforward and providing a universal language to communicate mathematical ideas.