Problem 21 - Prealgebra

Question

Describe each sequence using words and symbols. $$3,7,11,15, \dots$$

Step-by-Step Solution

Answer

The sequence is arithmetic, starting at 3, with a common difference of 4, expressed as \(a_n = 4n - 1\).

Step 1: Observe the Sequence

Look at the numbers in the sequence: 3, 7, 11, 15, and so on. Notice that each number is greater than the previous number and there is a regular pattern to how it increases.

Step 2: Determine the Pattern

Subtract two consecutive terms to see the pattern. For instance, \(7 - 3 = 4\), \(11 - 7 = 4\), and \(15 - 11 = 4\). The difference between consecutive terms is constant at 4.

Step 3: Define the Sequence Type

The sequence is an arithmetic sequence because it has a constant difference between consecutive terms. In this case, the common difference \(d\) is 4.

Step 4: Describe the Sequence Using Words

This sequence starts at 3 and each subsequent term is obtained by adding 4 to the previous term. This repeating pattern continues indefinitely.

Step 5: Express the Sequence Algebraically

In an arithmetic sequence, each term \(a_n\) is defined by the formula \(a_n = a_1 + (n-1) imes d\). Here, \(a_1 = 3\) and \(d = 4\), so the sequence can be expressed as \(a_n = 3 + (n-1) imes 4\).

Step 6: Simplify the Formula

Simplify the expression \(a_n = 3 + (n-1) imes 4\) to get a final formula for the sequence: \(a_n = 4n - 1\).

Key Concepts

Sequence Patterns

A sequence pattern is the visible order and arrangement of numbers within a sequence. When you observe the sequence

• 3
• 7
• 11
• 15

you can identify that each number is derived in a consistent manner from the previous one. Recognizing patterns involves identifying either an arithmetic progression, or another regular rule giving the sequence shape.

In our example, the sequence pattern becomes clear as the numbers increase by the same amount repeatedly. Every time, the next number seems a logical continuation, predictable once the pattern is evident. This predictability is what defines an arithmetic sequence; a consistent pattern of progression that can be captured by a regular formula.

Common Difference

The common difference is a fundamental feature of an arithmetic sequence. It refers to the constant amount by which each term in the series increases or decreases. In mathematical terms, subtract successive terms to confirm this difference.

For the sequence 3, 7, 11, 15:

• \[7 - 3 = 4\]
• \[11 - 7 = 4\]
• \[15 - 11 = 4\]

The difference is consistently 4, marking it as an arithmetic sequence. This constant is integral because it reveals how to transition from one term to the next, thus ensuring the reliability of the sequence's progression.

Algebraic Expression

An algebraic expression enables us to define a sequence distinctly using variables and operations. Such expressions provide a concise means of encapsulating the entire sequence conceptually.

For an arithmetic sequence, we express it through the formula:\[a_n = a_1 + (n-1) \times d\]where:

• \(a_n\) = the \(n^{th}\) term of the sequence
• \(a_1\) = the first term of the sequence
• \(d\) = the common difference

In our given sequence, the algebraic expression starts with a known initial term and implements the constant difference, thereby allowing us to determine any term solely by its position in the sequence.

Term Formula

The term formula in arithmetic sequences grants the ability to pinpoint any term within a sequence without listing all preceding elements. This is particularly useful for large numbers where listing becomes impractical. A term formula is integral for efficient problem-solving.

Given the sequence 3, 7, 11, 15, the formula to find the \(n^{th}\) term is:\[a_n = 4n - 1\]Here's how it's used:

• \(a_1 = 3\) confirms that the first term is indeed \(4 \times 1 - 1\).
• The 5th term, \(a_5\), can be calculated as \(4 \times 5 - 1 = 19\).

This approach simplifies direct computation of any term, demonstrating the power of understanding and applying sequence formulas.