5.1 Number Patterns & Sequences

5.1 Number Patterns & Sequences

Recognizing, extending, and formalizing patterns is a foundational pillar of elementary algebra. The Praxis Elementary Education: Mathematics Subtest (5003) requires candidates to analyze numerical and visual patterns, distinguish between linear and non-linear relationships, and derive explicit algebraic rules to find the $n$-th term of a sequence.

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1. Sequence Terminology & Foundations

A sequence is an ordered list of numbers that follows a specific mathematical rule. Each number in a sequence is called a term.

• Term Position ($n$): The index or placement of a term in the sequence (e.g., $n = 1$ for the 1st term, $n = 2$ for the 2nd term, $n = 50$ for the 50th term).
• Term Value ($a_n$): The numerical value located at position $n$. The symbol $a_1$ represents the initial term, $a_2$ the second term, and $a_n$ the general or $n$-th term.
• Finite vs. Infinite Sequences: A finite sequence has a fixed number of terms (e.g., $2, 4, 6, 8$), whereas an infinite sequence continues indefinitely (indicated by an ellipsis: $2, 4, 6, 8, \dots$).

Fundamental Sequence Types

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2. Arithmetic Sequences (Linear Patterns)

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. This constant value is called the common difference ($d$).

$d = a_n - a_{n-1}$

If $d > 0$, the sequence is increasing; if $d < 0$, the sequence is decreasing.

Deriving the Explicit Formula

To find the value of the 100th term, listing out all 100 terms is inefficient and prone to calculation errors. Instead, we use an explicit formula that links the term position $n$ directly to the term value $a_n$.

Starting with initial term $a_1$:

• $a_1 = a_1$
• $a_2 = a_1 + d$
• $a_3 = a_2 + d = a_1 + 2d$
• $a_4 = a_3 + d = a_1 + 3d$

Notice that to reach term $n$, we add the common difference $d$ exactly $(n - 1)$ times to the first term $a_1$.

$\text{Arithmetic Sequence Explicit Formula: } a_n = a_1 + (n - 1)d$

Step-by-Step Worked Example 1: Arithmetic Analysis

Problem: Consider the number sequence: $7, 12, 17, 22, 27, \dots$

1. Identify the common difference ($d$).
2. Write the explicit algebraic rule for the $n$-th term.
3. Calculate the 50th term ($a_{50}$).
4. Determine which term position has a value of $142$.

Solution:

• Step 1 (Find $d$): Subtract consecutive terms: $12 - 7 = 5$, $17 - 12 = 5$. Thus, $d = 5$.
• Step 2 (Write Explicit Rule): Substitute $a_1 = 7$ and $d = 5$ into the explicit formula: $a_n = 7 + (n - 1)5 = 7 + 5n - 5 = 5n + 2$
• Step 3 (Find 50th Term): Substitute $n = 50$ into the simplified explicit rule: $a_{50} = 5(50) + 2 = 250 + 2 = 252$
• Step 4 (Find Position for Value 142): Set $a_n = 142$ and solve for $n$: $5n + 2 = 142 \implies 5n = 140 \implies n = 28$ The 28th term of the sequence is $142$.

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3. Geometric Sequences (Exponential Patterns)

A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a constant non-zero value called the common ratio ($r$).

$r = \frac{a_n}{a_{n-1}}$

If $r > 1$, the sequence exhibits rapid exponential growth. If $0 < r < 1$, the sequence exhibits exponential decay.

$\text{Geometric Sequence Explicit Formula: } a_n = a_1 \cdot r^{n-1}$

Step-by-Step Worked Example 2: Geometric Analysis

Problem: Given the sequence $3, 6, 12, 24, 48, \dots$

1. Determine the common ratio ($r$).
2. Write the explicit formula for $a_n$.
3. Compute the 8th term ($a_8$).

Solution:

• Step 1 (Find $r$): Divide consecutive terms: $\frac{6}{3} = 2$, $\frac{12}{6} = 2$. Thus, $r = 2$.
• Step 2 (Explicit Formula): With $a_1 = 3$ and $r = 2$: $a_n = 3 \cdot 2^{n-1}$
• Step 3 (Find 8th Term): Substitute $n = 8$: $a_8 = 3 \cdot 2^{8-1} = 3 \cdot 2^7 = 3 \cdot 128 = 384$

Special Cases on the Exam:

• Decay ($r = \frac{1}{2}$): $64, 32, 16, 8, 4, \dots \implies a_n = 64 \cdot (\frac{1}{2})^{n-1}$
• Alternating signs ($r < 0$): $2, -6, 18, -54, 162, \dots \implies r = -3$

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4. Non-Linear & Special Sequences

Praxis 5003 frequently tests non-linear sequences that do not have a constant first difference or constant ratio.

A. Square Numbers ($a_n = n^2$)

The sequence of perfect squares is $1, 4, 9, 16, 25, 36, 49, \dots$

• First Differences: $+3, +5, +7, +9, +11, \dots$ (consecutive odd numbers).
• Second Differences: $+2, +2, +2, +2, \dots$ (a constant second difference indicates a quadratic sequence of degree 2).

B. Triangular Numbers ($T_n = \frac{n(n+1)}{2}$)

Triangular numbers represent objects that can form equilateral triangles: $1, 3, 6, 10, 15, 21, 28, \dots$

• Visual Construction: Row 1 has 1 dot ($1$), Row 2 adds 2 dots ($1+2=3$), Row 3 adds 3 dots ($3+3=6$), Row 4 adds 4 dots ($6+4=10$).
• Formula: The $n$-th triangular number is the sum of the first $n$ positive integers: $T_n = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$
• Example: The 8th triangular number is $T_8 = \frac{8(9)}{2} = \frac{72}{2} = 36$.

C. Fibonacci & Recursive Sequences

In a Fibonacci sequence, each term is the sum of the two preceding terms: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots$

• Recursive Definition: $a_1 = 1$, $a_2 = 1$, and $a_n = a_{n-1} + a_{n-2}$ for $n \ge 3$.
• Fibonacci Variations: A sequence might start with different initial values but follow the same additive rule. For instance, if $a_1 = 3$ and $a_2 = 5$, the sequence proceeds: $3, 5, 8, 13, 21, 34, \dots$

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5. Systematic Strategy for Finding the $n$-th Term Formula

When presented with a linear sequence on the Praxis exam, follow this foolproof 4-step method to write the explicit rule in $dn + c$ form:

1. Calculate Consecutive Differences: Subtract $a_1$ from $a_2$. If the difference is constant $d$, the rule is linear.
2. Assign the Coefficient: Set the coefficient of $n$ equal to the common difference $d$. The formula starts as $a_n = dn + c$.
3. Solve for Constant $c$: Evaluate the expression for $n = 1$ and set it equal to $a_1$: $d(1) + c = a_1 \implies c = a_1 - d$
4. Verify: Check the derived formula against $n = 2$ and $n = 3$.

Practice Application:

Find the formula for: $4, 9, 14, 19, 24, \dots$

• Step 1: First difference $d = 9 - 4 = 5$.
• Step 2: Form begins as $5n + c$.
• Step 3: For $n = 1$, $5(1) + c = 4 \implies 5 + c = 4 \implies c = -1$.
• Step 4: Explicit formula is $a_n = 5n - 1$. Verify for $n = 3$: $5(3) - 1 = 14$ (Correct!).

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6. Praxis Exam Traps & Pedagogical Recommendations

Common Candidate Pitfalls:

1. Confusing Position ($n$) and Value ($a_n$): Candidates often substitute the term value into $n$ instead of setting $a_n$ equal to the value when finding term placement.
2. Forgetting $(n-1)$ in Arithmetic Formulas: Writing $a_n = a_1 + nd$ instead of $a_1 + (n-1)d$ shifts the sequence by one position.
3. Assuming All Increasing Patterns are Arithmetic: Always check at least two pairs of consecutive terms to verify whether the difference or ratio is truly constant.

Elementary Classroom Strategy: When teaching patterns to elementary students, transition from concrete manipulatives (color tiles, pattern blocks) to pictorial representations (dot diagrams, t-charts), and finally to abstract algebraic formulas. This concrete-representational-abstract (CRA) sequence builds deep conceptual understanding of functional growth.

Shape Patterns and Corresponding Terms

Growing shape patterns

Elementary items often show a sequence of figures (for example, toothpick squares or dotted triangles) and ask for the next figure or a rule for the $n$th figure. Count a measurable attribute—sides, toothpicks, tiles—and build a table relating figure number $n$ to that count. Many grow arithmetically (constant first difference); some grow by a quadratic rule (constant second difference), such as square numbers.

Corresponding terms of two numerical patterns

ETS also asks you to relate two patterns generated by different rules. Example: Pattern A starts at 0 and adds 3; Pattern B starts at 0 and adds 6. Corresponding terms are $(0,0)$, $(3,6)$, $(6,12)$, … and each B term is twice the matching A term. A function-table rule such as “B = 2A” captures the relationship between corresponding terms.