Problem 18 - Prealgebra

Question

Describe each sequence using words and symbols. $$3,5,7,9, \dots$$

Step-by-Step Solution

Answer

The sequence is arithmetic with the formula: \( a_n = 2n + 1 \).

Step 1: Identify the Pattern

Observe the sequence provided: 3, 5, 7, 9, ... Notice that each term increases by 2 from the previous term.

Step 2: Express the Pattern Verbally

The sequence starts at 3 and increases by 2 with each successive term. Each term is the sum of the previous term and the common difference, 2.

Step 3: Find the General Formula

Recognize this as an arithmetic sequence where the first term (\( a_1 \)) is 3, and the common difference (\( d \)) is 2. The general form for an arithmetic sequence is given by \( a_n = a_1 + (n-1)d \).

Step 4: Simplify the Formula

Substitute the values for the first term and the common difference into the general formula: \( a_n = 3 + (n-1) imes 2 \). Simplify this to \( a_n = 3 + 2n - 2 \), which further simplifies to \( a_n = 2n + 1 \).

Key Concepts

Pattern Recognition

In the world of mathematics, identifying the repetition or regular changes in a sequence is known as **pattern recognition**. This process allows us to understand how sequences progress. For example, consider the sequence 3, 5, 7, 9, ... First, check how each number relates to the next. Notice that each number appears to increase by the same amount, which is 2, revealing a simple pattern. Recognizing this pattern is crucial, as it tells us that the progression between numbers is consistent and predictable.

General Formula

Once we have identified a consistent pattern in a sequence, we can derive a **general formula**. This formula serves as a blueprint that represents the entire sequence in mathematical terms. For an arithmetic sequence, where numbers increase by a fixed number (the common difference), the general formula is:\[ a_n = a_1 + (n-1) \cdot d \]Here, \( a_1 \) is the first term of the sequence, \( d \) is the common difference, and \( n \) represents the term number. This formula simplifies predicting any term in the sequence without listing all previous terms. The specific formula for our sequence 3, 5, 7, 9, ... is:\[ a_n = 3 + (n-1) \cdot 2 \]Simplifying, we get \( a_n = 2n + 1 \).

Common Difference

The **common difference** is a key feature of an arithmetic sequence, guiding how the sequence grows. It is the amount added to each term to get to the next. In our example sequence 3, 5, 7, 9, ..., it's clear that each term increases by 2.

• Simply put, the common difference is the hills and valleys of our sequence's journey.
• Using it allows us to create predictable sequences smoothly.
• In formula notation, the common difference is labeled as \( d \).

In particular, for this sequence, \( d = 2 \). Knowing the common difference helps us quickly determine other terms in the sequence by repetitive addition.

Sequence Description

To offer a comprehensive **sequence description**, both numerical and verbal explanations are helpful. The sequence we are exploring, beginning with 3, and continuing as 3, 5, 7, 9, ..., is simple yet structured.

• Numerically, it's an arithmetic sequence starting at 3, adding 2 at each step.
• Verbally, you would describe it as: a sequence where each subsequent term follows from adding 2 to the previous term.

This dual description allows learners to appreciate both the formulaic and narrative aspects of sequences, giving a clearer picture of the entire pattern. The sequence is succinctly described using formulas, yet explained in words for better clarity. This twin approach enhances understanding and application of arithmetic sequences in various mathematical contexts.