Algebra Flashcards: Sequences As Functions And Recursion

What this deck covers

This deck focuses on Sequences As Functions And Recursion, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Identify whether $a_n=2n-1$ (for $n\ge 1$) is explicit or recursive.

Answer: Explicit. The formula directly gives $a_n$ in terms of $n$.

Flashcard 2: What is the range of a sequence?

Answer: The set of term values $\{f(n)\}$ produced by allowed integer $n$. The range contains all possible output values of the function.

Flashcard 3: What is $a_4$ for the explicit sequence $a_n=2^n$ with domain $n\ge 0$?

Answer: $a_4=16$. Substituting $n=4$ into the explicit formula: $a_4=2^4=16$.

Flashcard 4: What is the output of a sequence when the input is not an integer in its domain?

Answer: It is undefined (not part of the sequence). Sequences only exist at integer inputs in their domain.

Flashcard 5: What does the notation $f(n)$ represent when $f$ is a sequence?

Answer: The output (term) of the sequence at integer input $n$. The function notation shows the sequence value at position $n$.

Flashcard 6: What is the recursive definition of the Fibonacci sequence given in the standard?

Answer: $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$ for $n\ge 1$. This is the standard recursive definition from CCSS.F-IF.3.

Flashcard 7: What is $a_6$ for the explicit sequence $a_n=n^2$?

Answer: $a_6=36$. Substituting $n=6$ into the explicit formula: $a_6=6^2=36$.

Flashcard 8: What is the input variable for a sequence written as $a_n$?

Answer: The index $n$, which is an integer in the domain. The subscript $n$ represents the independent variable (input).

Flashcard 9: Which set is a valid domain for a sequence: $\mathbb{R}$ or $\{1,2,3,\dots\}$?

Answer: $\{1,2,3,\dots\}$. Sequences must have integer domains, not all real numbers.

Flashcard 10: What does it mean for a sequence to be defined explicitly?

Answer: A formula gives $a_n$ directly in terms of $n$. No need for previous terms; just substitute $n$ into the formula.

Flashcard 11: Find $a_3$ if $a_0=4$ and $a_n=a_{n-1}-n$ for $n\ge 1$.

Answer: $a_3=-2$. $a_2=a_1-2=3-2=1$, $a_3=a_2-3=1-3=-2$.

Flashcard 12: Find $a_2$ if $a_0=5$ and $a_n=2a_{n-1}$ for $n\ge 1$.

Answer: $a_2=20$. $a_1=2(5)=10$, $a_2=2(10)=20$.

Flashcard 13: What is $f(3)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?

Answer: $f(3)=3$. Using $f(2)=2$ and $f(1)=1$: $f(3)=f(2)+f(1)=2+1=3$.

Flashcard 14: What is a recursion rule for a sequence?

Answer: An equation such as $a_n=\text{(expression using earlier terms)}$. The rule shows how to calculate each term from earlier ones.

Flashcard 15: Find $a_2$ if $a_0=4$ and $a_n=a_{n-1}-n$ for $n\ge 1$.

Answer: $a_2=1$. $a_1=a_0-1=4-1=3$, $a_2=a_1-2=3-2=1$.

Flashcard 16: What is the ordered-pair representation of a sequence term $a_n$ on a graph?

Answer: The point $(n,a_n)$. The $x$-coordinate is $n$, the $y$-coordinate is the term value.

Flashcard 17: What does it mean for a sequence to be defined recursively?

Answer: Terms are defined using previous term(s) plus initial condition(s). Each term depends on one or more preceding terms.

Flashcard 18: Find $a_3$ if $a_1=-1$ and $a_n=a_{n-1}^2$ for $n\ge 2$.

Answer: $a_3=1$. Squaring the previous term: $a_3=(1)^2=1$.

Flashcard 19: What is $f(2)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?

Answer: $f(2)=2$. Using $f(1)=1$ and $f(0)=1$: $f(2)=f(1)+f(0)=1+1=2$.

Flashcard 20: Identify the domain for the sequence points $(1,4)$, $(2,7)$, $(3,10)$, $(4,13)$.

Answer: $\{1,2,3,4\}$. The domain consists of all $x$-coordinates of the points.

Flashcard 21: Identify the range for the sequence points $(1,4)$, $(2,7)$, $(3,10)$, $(4,13)$.

Answer: $\{4,7,10,13\}$. The range consists of all $y$-coordinates of the points.

Flashcard 22: What is $a_0$ for the explicit sequence $a_n=2^n$ with domain $n\ge 0$?

Answer: $a_0=1$. Substituting $n=0$ into the explicit formula: $a_0=2^0=1$.

Flashcard 23: What is the value of $a_3$ if the sequence is given by points $(0,2)$, $(1,5)$, $(2,8)$, $(3,11)$?

Answer: $a_3=11$. The point $(3,11)$ shows that $a_3=11$.

Flashcard 24: What is $a_1$ for the explicit sequence $a_n=2^n$ with domain $n\ge 0$?

Answer: $a_1=2$. Substituting $n=1$ into the explicit formula: $a_1=2^1=2$.

Flashcard 25: What is the correct interpretation of $f: \{0,1,2,\dots\}\to\mathbb{R}$?

Answer: $f$ assigns a real value to each integer input $n\ge 0$. Function notation shows integer inputs map to real number outputs.

Flashcard 26: Which representation best matches a sequence: discrete points or a continuous curve?

Answer: Discrete points $(n,a_n)$ at integer $n$. Sequences have gaps between integer inputs, unlike continuous functions.

Flashcard 27: What is the definition of a sequence as a function in Algebra 1?

Answer: A function with domain a subset of the integers, usually $\{0,1,2,\dots\}$. The domain must be integers, not continuous real numbers.

Flashcard 28: What is required to define a recursive sequence that uses $a_{n-1}$ only?

Answer: One initial condition (for example, $a_1$) and the recursion rule. One previous term requires one starting value to begin.

Flashcard 29: Find $a_3$ if $a_0=2$, $a_1=3$, and $a_n=a_{n-1}+a_{n-2}$ for $n\ge 2$.

Answer: $a_3=8$. Adding the two previous terms: $a_3=a_2+a_1=5+3=8$.

Flashcard 30: Find $a_3$ if $a_1=10$ and $a_n=a_{n-1}-4$ for $n\ge 2$.

Answer: $a_3=2$. $a_2=10-4=6$, $a_3=6-4=2$.

Flashcard 31: Find $a_3$ if $a_1=2$ and $a_n=\frac{1}{2}a_{n-1}$ for $n\ge 2$.

Answer: $a_3=\frac{1}{2}$. $a_2=\frac{1}{2}(2)=1$, $a_3=\frac{1}{2}(1)=\frac{1}{2}$.

Flashcard 32: What is $a_5$ for the explicit sequence $a_n=3n$?

Answer: $a_5=15$. Substituting $n=5$ into the explicit formula: $a_5=3(5)=15$.

Flashcard 33: Identify whether $a_1=1$ and $a_n=a_{n-1}+3$ (for $n\ge 2$) is explicit or recursive.

Answer: Recursive. Each term is defined using the previous term.

Flashcard 34: Identify the domain of Fibonacci as defined by $f(0)$, $f(1)$, and $f(n+1)$ for $n\ge 1$.

Answer: All integers $n\ge 0$. Initial conditions define $f(0)$ and $f(1)$, rule applies for $n\ge 1$.

Flashcard 35: What is the index of the term $11$ in the sequence points $(0,2)$, $(1,5)$, $(2,8)$, $(3,11)$?

Answer: $n=3$. The point $(3,11)$ shows that term $11$ occurs at index $3$.

Flashcard 36: What is $f(5)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?

Answer: $f(5)=8$. Using $f(4)=5$ and $f(3)=3$: $f(5)=f(4)+f(3)=5+3=8$.

Flashcard 37: Find $a_2$ if $a_1=-1$ and $a_n=a_{n-1}^2$ for $n\ge 2$.

Answer: $a_2=1$. Squaring the previous term: $a_2=(-1)^2=1$.

Flashcard 38: What is $f(3)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?

Answer: $f(3)=3$. Using $f(2)=2$ and $f(1)=1$: $f(3)=f(2)+f(1)=2+1=3$.

Flashcard 39: What is $f(4)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?

Answer: $f(4)=5$. Using $f(3)=3$ and $f(2)=2$: $f(4)=f(3)+f(2)=3+2=5$.

Flashcard 40: What does the condition $n\ge 1$ mean in a recursion like $f(n+1)=f(n)+f(n-1)$?

Answer: The rule applies only for integer inputs $n$ starting at $1$. The recursion needs $n\ge 1$ to ensure both $f(n)$ and $f(n-1)$ exist.

Flashcard 41: Find $a_2$ if $a_0=2$, $a_1=3$, and $a_n=a_{n-1}+a_{n-2}$ for $n\ge 2$.

Answer: $a_2=5$. Adding the two previous terms: $a_2=a_1+a_0=3+2=5$.

Flashcard 42: Identify the missing initial condition needed for $a_n=2a_{n-1}+1$ to define a sequence.

Answer: A starting value such as $a_0$ or $a_1$. Recursive definitions require at least one starting value.

Flashcard 43: Choose the correct domain statement for $a_n=\frac{1}{n}$ when written as a sequence.

Answer: Integers $n\ge 1$. Sequences require integer domains, not all real numbers.

Flashcard 44: What is the domain of a typical sequence written as $a_n$ for $n=1,2,3,\dots$?

Answer: The positive integers $\{1,2,3,\dots\}$. Starting at $n=1$ gives positive integers as the domain.

Flashcard 45: What is $a_3$ for the explicit sequence $a_n=2^n$ with domain $n\ge 0$?

Answer: $a_3=8$. Substituting $n=3$ into the explicit formula: $a_3=2^3=8$.

Flashcard 46: What is required to define a recursive sequence that uses $a_{n-1}$ and $a_{n-2}$?

Answer: Two initial conditions (for example, $a_0$ and $a_1$) and the rule. Two previous terms require two starting values to begin.

Flashcard 47: What is the domain of a sequence defined for $n=0,1,2,\dots$?

Answer: The nonnegative integers $\{0,1,2,\dots\}$. Starting at $n=0$ includes zero in the domain.

Flashcard 48: What is the key difference between a sequence and a continuous function graph?

Answer: A sequence has discrete integer inputs, not all real $x$-values. Sequences only exist at integer points, not between them.

Flashcard 49: What is the smallest number of initial conditions needed for $a_n=a_{n-1}+a_{n-2}$?

Answer: Two initial conditions. The rule uses two previous terms, so two starting values are needed.

Flashcard 50: Find $a_3$ if $a_0=1$ and $a_n=3a_{n-1}$ for $n\ge 1$.

Answer: $a_3=27$. $a_1=3(1)=3$, $a_2=3(3)=9$, $a_3=3(9)=27$.

Flashcard 51: What is $f(4)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?

Answer: $f(4)=5$. Using $f(3)=3$ and $f(2)=2$: $f(4)=f(3)+f(2)=3+2=5$.

Flashcard 52: What does the notation $a_n$ mean in a sequence?

Answer: The $n$th term of the sequence. Subscript notation indicates the term's position in the sequence.

Flashcard 53: Find $a_4$ if $a_1=3$ and $a_n=a_{n-1}+2$ for $n\ge 2$.

Answer: $a_4=9$. $a_2=3+2=5$, $a_3=5+2=7$, $a_4=7+2=9$.

Flashcard 54: What is an initial condition in a recursive sequence?

Answer: Starting value(s) like $a_1$ or $a_0$ needed to generate later terms. Without initial values, the recursive pattern cannot begin.