8.1 Arithmetic and Geometric Sequences

8.1 Arithmetic and Geometric Sequences

Quick Answer: A sequence is a function whose domain consists of consecutive integers (typically $n \in {1, 2, 3, \dots}$). An arithmetic sequence adds a constant common difference $d$ to each term ($a_n = a_1 + (n-1)d$), exhibiting discrete linear growth with slope $m = d$. A geometric sequence multiplies each term by a constant common ratio $r$ ($a_n = a_1 r^{n-1}$), exhibiting discrete exponential growth with base $b = r$. Explicit formulas calculate any term directly from its index $n$, while recursive formulas specify an initial term $a_1$ and define $a_n$ from $a_{n-1}$.

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1. Sequences as Functions on Discrete Integer Domains (F-BF.1a, F-BF.2)

In elementary algebra, functions are typically introduced over continuous real domains: a linear function $f(x) = 3x + 2$ or an exponential function $g(x) = 5(2)^x$ accepts any real input $x \in \mathbb{R}$, yielding unbroken, continuous graphical curves. Under the New York State Next Generation Learning Standards, students transition to analyzing sequences through the formal lens of function theory.

The Formal Definition of a Sequence

A sequence is a function $f$ whose domain is a subset of the integers—almost universally the set of positive integers (natural numbers):

$\text{Domain: } \mathbb{N} = \{1, 2, 3, 4, \dots, n, \dots\}$

Rather than writing function notation $f(n)$, mathematicians use subscript notation $a_n$ to designate the output value corresponding to the input integer $n$:

$f(1) = a_1, \quad f(2) = a_2, \quad f(3) = a_3, \quad \dots, \quad f(n) = a_n$

• The input integer $n$ is called the term index or position.
• The output value $a_n$ is called the $n$-th term or general term.
• The first output $a_1$ is called the initial term.

Because the domain contains only isolated integers, the graph of a sequence in the coordinate plane consists strictly of discrete, disconnected points $(1, a_1), (2, a_2), (3, a_3), \dots$. It is mathematically incorrect to connect the plotted points with a solid line or curve, because values such as $a_{2.5}$ or $a_{\sqrt{2}}$ do not exist.

Explicit vs. Recursive Representations

Every sequence can be described using two distinct mathematical frameworks:

1. Explicit Formulas: Express the term value $a_n$ solely in terms of its position index $n$. This allows direct computation of any arbitrary term without prior knowledge of earlier terms. For example, to find the 100th term, one simply evaluates $a_{100}$.
2. Recursive Formulas: Express the term value $a_n$ as a function of the immediately preceding term $a_{n-1}$ (or multiple preceding terms). A complete recursive definition strictly requires two components: an initial condition (base case) defining $a_1$, and a recurrence relation governing $a_n$ for all integers $n \ge 2$.

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2. Arithmetic Sequences: Differences, Formulas, and Linear Models

An arithmetic sequence is a sequence in which the difference between any two consecutive terms is constant. This constant value is designated the common difference, symbolized by $d$.

Calculating the Common Difference

To determine whether a sequence is arithmetic, compute the difference between successive terms:

$d = a_n - a_{n-1} = a_2 - a_1 = a_3 - a_2 = a_4 - a_3$

• If $d > 0$, the sequence is strictly increasing.
• If $d < 0$, the sequence is strictly decreasing.
• If $d = 0$, the sequence is constant.

Deriving the Explicit Formula

Tracing the sequence term-by-term reveals a clear structural pattern:

a_1 &= a_1 \\ a_2 &= a_1 + d \\ a_3 &= a_2 + d = (a_1 + d) + d = a_1 + 2d \\ a_4 &= a_3 + d = (a_1 + 2d) + d = a_1 + 3d \\ &\;\;\vdots \\ a_n &= a_1 + (n - 1)d \end{aligned}$$ Notice that to advance from the first term $a_1$ to the $n$-th term $a_n$, exactly $(n - 1)$ additions of the common difference $d$ are executed. This explicit formula is explicitly provided on the **NYSED Next Generation Mathematics Reference Sheet**. ### The Recursive Definition The recursive formula translates the stepwise addition of $d$ directly into mathematical notation: $$\begin{cases} a_1 = k \\ a_n = a_{n-1} + d, & \text{for } n \ge 2 \end{cases}$$ > [!IMPORTANT] > **The Base Case Rule on Regents Exams**: When asked to write a recursive formula on Part II or Part III constructed-response items, failing to state $a_1 = k$ results in an automatic 1-credit penalty. A recurrence relation $a_n = a_{n-1} + d$ alone is incomplete because an infinite number of sequences share the same common difference but start at different initial values. ### The Connection Between Arithmetic Sequences and Linear Functions Expanding the explicit formula highlights its underlying algebraic identity with linear functions: $$a_n = a_1 + (n - 1)d = a_1 + dn - d = dn + (a_1 - d)$$ Comparing this structure to slope-intercept form $f(x) = mx + b$: - The common difference $d$ corresponds exactly to the **constant rate of change (slope $m$)**. - The constant term $(a_1 - d)$ corresponds to the **vertical intercept (the hypothetical 0th term $a_0$)**. | Mathematical Attribute | Continuous Linear Function $f(x) = mx + b$ | Discrete Arithmetic Sequence $a_n = dn + (a_1 - d)$ | | :--- | :--- | :--- | | **Domain** | All real numbers: $x \in \mathbb{R}$ | Positive integers: $n \in \{1, 2, 3, \dots\}$ | | **Rate of Change** | Constant slope: $m = \frac{\Delta y}{\Delta x}$ | Common difference: $d = a_n - a_{n-1}$ | | **Vertical Intercept** | Value at $x = 0$: $f(0) = b$ | Hypothetical 0th term: $a_0 = a_1 - d$ | | **Graphical Appearance** | Unbroken continuous line | Collinear set of isolated points | --- ## 3. Geometric Sequences: Ratios, Formulas, and Exponential Models A **geometric sequence** is a sequence in which the ratio between any two consecutive terms is constant. This multiplier is designated the **common ratio**, symbolized by $r$. ### Calculating the Common Ratio To determine whether a sequence is geometric, divide each term by its immediate predecessor: $$r = \frac{a_n}{a_{n-1}} = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} \quad (a_{n-1} \neq 0)$$ - If $r > 1$ and $a_1 > 0$, terms grow exponentially without bound. - If $0 < r < 1$ and $a_1 > 0$, terms decay exponentially toward 0. - If $r < 0$, terms **alternate in sign** between positive and negative values (an oscillating sequence). ### Deriving the Explicit Formula Tracing the multiplication term-by-term reveals the exponential progression: $$\begin{aligned} a_1 &= a_1 \\ a_2 &= a_1 \cdot r \\ a_3 &= a_2 \cdot r = (a_1 \cdot r) \cdot r = a_1 r^2 \\ a_4 &= a_3 \cdot r = (a_1 r^2) \cdot r = a_1 r^3 \\ &\;\;\vdots \\ a_n &= a_1 r^{n - 1} \end{aligned}$$ To advance from $a_1$ to $a_n$, the initial value is multiplied by $r$ exactly $(n - 1)$ times. This explicit formula is provided on the official **NYSED Reference Sheet**. ### The Recursive Definition $$\begin{cases} a_1 = k \\ a_n = r \cdot a_{n-1}, & \text{for } n \ge 2 \end{cases}$$ ### The Connection Between Geometric Sequences and Exponential Functions Rewriting the explicit formula using exponent rules demonstrates its structural identity with exponential models: $$a_n = a_1 r^{n - 1} = a_1 \cdot r^n \cdot r^{-1} = \left(\frac{a_1}{r}\right) r^n$$ Comparing this to standard exponential form $f(x) = a \cdot b^x$: - The common ratio $r$ corresponds to the **growth/decay factor (base $b$)**. - The leading coefficient $\frac{a_1}{r}$ represents the **initial value $f(0) = a_0$**. > [!NOTE] > **Alternating Sequences vs. Real Exponential Curves**: Continuous exponential functions $f(x) = a \cdot b^x$ require the base $b > 0$ because negative bases raised to fractional powers (such as $(-4)^{1/2} = 2i$) produce non-real complex numbers. In contrast, geometric sequences restrict their domain to integer powers $n \in \mathbb{Z}^+$. Therefore, geometric sequences **permit negative common ratios** $r < 0$, which generate alternating sign sequences such as $3, -6, 12, -24, 48, \dots$. --- ## 4. Comprehensive Structural Comparison | Sequence Property | Arithmetic Sequence | Geometric Sequence | | :--- | :--- | :--- | | **Defining Operation** | Successive addition of constant $d$ | Successive multiplication by constant $r$ | | **Identifying Test** | $a_2 - a_1 = a_3 - a_2 = d$ | $\frac{a_2}{a_1} = \frac{a_3}{a_2} = r$ | | **Explicit Formula (Ref. Sheet)** | $a_n = a_1 + (n - 1)d$ | $a_n = a_1 r^{n - 1}$ | | **Recursive Definition** | $a_1 = k, \; a_n = a_{n-1} + d$ | $a_1 = k, \; a_n = r \cdot a_{n-1}$ | | **Underlying Function Family** | Linear: $f(x) = mx + b$ | Exponential: $f(x) = a \cdot b^x$ | | **Growth Parameter** | Slope: $m = d$ | Base multiplier: $b = r$ | | **Hypothetical 0th Term** | $a_0 = a_1 - d$ | $a_0 = \frac{a_1}{r}$ | | **Behavior when $d < 0$ or $0 < r < 1$** | Decreases linearly toward $-\infty$ | Decays exponentially toward 0 | --- ## 5. Worked Problem 1: Converting Between Recursive and Explicit Forms **Problem:** A sequence is defined recursively by $a_1 = 19$ and $a_n = a_{n-1} - 3.5$ for $n \ge 2$. Write an explicit formula for the sequence, calculate the value of $a_{32}$, and determine the linear function $f(n)$ that models this sequence. - **Step 1: Identify the sequence type and parameters.** The recurrence relation subtracts a constant $3.5$ from each term, indicating an arithmetic sequence with: $$a_1 = 19, \quad d = -3.5$$ - **Step 2: Construct the explicit formula.** Using the reference sheet formula $a_n = a_1 + (n - 1)d$: $$a_n = 19 + (n - 1)(-3.5)$$ - **Step 3: Simplify into slope-intercept form.** $$a_n = 19 - 3.5n + 3.5 = -3.5n + 22.5$$ The corresponding linear function is $f(n) = -3.5n + 22.5$ for $n \in \mathbb{N}$. - **Step 4: Compute the 32nd term.** Substitute $n = 32$ into the explicit formula: $$a_{32} = -3.5(32) + 22.5 = -112 + 22.5 = -89.5$$ --- ## 6. Worked Problem 2: Finding Missing Terms Between Non-Consecutive Geometric Terms **Problem:** In a geometric sequence of positive real numbers, the third term is $a_3 = 24$ and the seventh term is $a_7 = 384$. Algebraically determine the common ratio $r$, the initial term $a_1$, write the explicit formula, and state the values of the intermediate missing terms $a_4, a_5,$ and $a_6$. - **Step 1: Express both terms using the explicit geometric formula.** Using $a_n = a_1 r^{n-1}$: $$a_3 = a_1 r^{3 - 1} = a_1 r^2 = 24$$ $$a_7 = a_1 r^{7 - 1} = a_1 r^6 = 384$$ - **Step 2: Solve for $r$ by forming the quotient of the equations.** $$\frac{a_7}{a_3} = \frac{a_1 r^6}{a_1 r^2} = \frac{384}{24}$$ $$r^{6 - 2} = r^4 = 16$$ - **Step 3: Extract the fourth root.** $$r = \pm \sqrt[4]{16} = \pm 2$$ Because the problem prompt explicitly specifies that the sequence consists of positive real numbers, reject the negative root: $$r = 2$$ - **Step 4: Calculate the first term $a_1$.** Substitute $r = 2$ back into the equation for $a_3$: $$a_1 (2)^2 = 24 \implies 4a_1 = 24 \implies a_1 = 6$$ - **Step 5: State the explicit formula and calculate missing terms.** $$a_n = 6(2)^{n - 1}$$ Evaluating consecutive terms: $$a_4 = 6(2)^3 = 48, \quad a_5 = 6(2)^4 = 96, \quad a_6 = 6(2)^5 = 192$$ --- ## 7. Worked Problem 3: Resolving Non-Consecutive Arithmetic Terms **Problem:** An arithmetic sequence has a 4th term of $a_4 = -7$ and an 11th term of $a_{11} = 35$. Determine the common difference $d$, the first term $a_1$, and the term index $n$ for which $a_n = 101$. - **Step 1: Set up equations for both terms.** $$a_4 = a_1 + 3d = -7$$ $$a_{11} = a_1 + 10d = 35$$ - **Step 2: Subtract the equations to eliminate $a_1$.** $$(a_1 + 10d) - (a_1 + 3d) = 35 - (-7)$$ $$7d = 42 \implies d = 6$$ - **Step 3: Solve for $a_1$.** $$a_1 + 3(6) = -7 \implies a_1 + 18 = -7 \implies a_1 = -25$$ - **Step 4: Determine the index $n$ where $a_n = 101$.** $$101 = -25 + (n - 1)6$$ $$126 = 6(n - 1) \implies 21 = n - 1 \implies n = 22$$ The 22nd term of the sequence is $101$. --- ## 8. Common Regents Pitfalls and Exam Strategies - **Pitfall 1: Index Off-by-One Exponent Error**: A frequent error on Part I items is confusing $r^{n-1}$ with $r^n$. In $a_n = a_1 r^{n-1}$, the exponent is $n - 1$ because the first term already has zero factors of $r$ ($a_1 = a_1 r^0$). Writing $a_n = a_1 r^n$ shifts the entire sequence by one index, making $a_1$ equal to $a_1 r$, which is incorrect. - **Pitfall 2: Forgetting the Base Case in Recursive Definitions**: When writing a recursive formula on Regents constructed-response items, you must explicitly state $a_1 = k$. Omitting $a_1$ forfeits 1 credit out of 2. - **Pitfall 3: Reversing the Subtraction or Division Order**: When computing common difference $d$ or common ratio $r$, always take the later term minus (or divided by) the earlier term: $d = a_2 - a_1$, never $a_1 - a_2$. - **Pitfall 4: Overlooking Dual Signs on Even Roots**: When solving $r^{k-j} = C$ where $k - j$ is even, $r = \pm \sqrt[k-j]{C}$. Unless the prompt restricts terms to positive real numbers, both positive and negative values for $r$ are mathematically valid.