Algebra 2 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 2.

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: What is the ordered-pair form for the $n$-th term of a sequence $f(n)$ on a graph?

Answer: The point $(n,f(n))$. Standard coordinate form with $n$ as input and $f(n)$ as output.

Flashcard 2: What is the output of the sequence defined by $f(n)=\frac{n-1}{2}$ at $n=7$?

Answer: $f(7)=3$. Substitute $n=7$: $f(7)=\frac{7-1}{2}=\frac{6}{2}=3$.

Flashcard 3: Identify the domain for a sequence defined by $f(n)=\frac{1}{n}$ with the restriction $n\ge 1$.

Answer: $\{1,2,3,\dots\}$. Domain excludes 0 to avoid division by zero in $\frac{1}{n}$.

Flashcard 4: What is the value of $f(3)$ if $f(0)=2$, $f(1)=5$, and $f(n+1)=f(n)+f(n-1)$ for $n\ge 1$?

Answer: $f(3)=12$. Apply recursion with $n=2$: $f(3)=f(2)+f(1)=7+5=12$.

Flashcard 5: What is the value of $f(4)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?

Answer: $f(4)=5$. Apply recursion: $f(4)=f(3)+f(2)=3+2=5$.

Flashcard 6: What is an explicit formula for a sequence?

Answer: A formula that gives $f(n)$ directly in terms of $n$. Direct formulas compute terms without needing previous terms.

Flashcard 7: What is the value of $f(3)$ if $f(n)=f(n-1)+4$ with $f(1)=2$?

Answer: $f(3)=10$. Use recursion: $f(2)=2+4=6$, $f(3)=6+4=10$.

Flashcard 8: What is the value of $f(7)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?

Answer: $f(7)=21$. Apply recursion: $f(7)=f(6)+f(5)=13+8=21$.

Flashcard 9: Identify whether $f(n)=\frac{1}{n}$ is defined at $n=0$ when the domain is $\{0,1,2,\dots\}$.

Answer: No, because $\frac{1}{0}$ is undefined. Division by zero makes the function undefined at $n=0$.

Flashcard 10: What is the $n$th term notation for the sequence values $f(1),f(2),f(3),\dots$?

Answer: The list of outputs indexed by integers: $f(1),f(2),f(3),\dots$. Function notation creates an indexed list of sequence outputs.

Flashcard 11: What is the value of $f(2)$ if $f(0)=2$, $f(1)=5$, and $f(n+1)=f(n)+f(n-1)$ for $n\ge 1$?

Answer: $f(2)=7$. Apply recursion with $n=1$: $f(2)=f(1)+f(0)=5+2=7$.

Flashcard 12: What is the output of the sequence defined by $f(n)=3\cdot 2^n$ at $n=3$?

Answer: $f(3)=24$. Substitute $n=3$: $f(3)=3\cdot 2^3=3\cdot 8=24$.

Flashcard 13: Which statement correctly describes the range of a sequence as a function?

Answer: The set of all outputs $f(n)$ for integer inputs in the domain. Range consists of all possible function outputs from the domain.

Flashcard 14: What does it mean to say a sequence is a function but not a typical real function graph?

Answer: It is defined only at integer inputs, so its graph is discrete points. Sequences are discrete functions with gaps between integer points.

Flashcard 15: What is the value of $f(6)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?

Answer: $f(6)=13$. Apply recursion: $f(6)=f(5)+f(4)=8+5=13$.

Flashcard 16: Which set best represents the domain of a sequence defined for $n\ge 1$?

Answer: $\{1,2,3,\dots\}$. This set includes all positive integers starting from 1.

Flashcard 17: Identify the correct interpretation of $f(n+1)=f(n)+f(n-1)$ in words.

Answer: Each term equals the sum of the previous two terms. This describes the Fibonacci-type addition pattern in words.

Flashcard 18: What is the definition of a sequence written as a function $f(n)$?

Answer: A function with domain in integers that assigns each $n$ a value $f(n)$. This defines sequences using function notation with integer domains.

Flashcard 19: What is a second-order recursion for a sequence?

Answer: A recursion that uses two previous terms, such as $f(n+1)=f(n)+f(n-1)$. Second-order means each term depends on exactly two previous terms.

Flashcard 20: What is the correct input set for a sequence term labeled $f(12)$?

Answer: The input is the integer $n=12$. The subscript 12 indicates the integer input to the sequence function.

Flashcard 21: What two parts must a recursive sequence definition include to determine all terms?

Answer: Initial condition(s) and a recursion rule. Both components are essential for uniquely determining the sequence.

Flashcard 22: What is the output $f(4)$ if $f(0)=4$, $f(1)=7$, and $f(n)=f(n-1)+f(n-2)$ for $n\ge 2$?

Answer: $f(4)=29$. Apply recursion: $f(4)=f(3)+f(2)=18+11=29$.

Flashcard 23: What is the meaning of the index $n$ in a sequence written $f(n)$?

Answer: The input that indicates position in the sequence. Index $n$ specifies which term in the sequence ordering.

Flashcard 24: Identify whether the relation $f(n)=\pm n$ on $\{1,2,3\}$ is a function.

Answer: Not a function, because one input $n$ can have two outputs. Functions require exactly one output per input, not multiple outputs.

Flashcard 25: Which option correctly states why a recursion alone does not define a unique sequence?

Answer: Without initial condition(s), many sequences satisfy the same recursion. Initial conditions distinguish between different sequences with same recursion.

Flashcard 26: In the Fibonacci definition $f(n+1)=f(n)+f(n-1)$ for $n\ge 1$, what is the smallest $n$ allowed?

Answer: $n=1$. The recursion rule applies starting from $n=1$ with given conditions.

Flashcard 27: Identify the correct domain description for the Fibonacci sequence defined by $f(0)=f(1)=1$.

Answer: All integers $n$ with $n\ge 0$. Domain starts at 0 since both $f(0)$ and $f(1)$ are defined.

Flashcard 28: What is a recursive definition of a sequence?

Answer: A definition using initial term(s) and a rule relating terms to earlier terms. Recursive definitions build terms from earlier computed values.

Flashcard 29: What is the value of $f(9)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?

Answer: $f(9)=55$. Apply recursion: $f(9)=f(8)+f(7)=34+21=55$.

Flashcard 30: Which option is the correct graph description for a sequence $f(n)$: points or a continuous curve?

Answer: Points only, at integer $n$ values. Sequences have discrete domains, creating point graphs not curves.

Flashcard 31: What is the value of $f(5)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?

Answer: $f(5)=8$. Apply recursion: $f(5)=f(4)+f(3)=5+3=8$.

Flashcard 32: What is the next term after $13$ in the Fibonacci sequence starting $1,1,2,3,5,8,13$?

Answer: $21$. Fibonacci rule: add the two previous terms $13+8=21$.

Flashcard 33: What is the notation $a_n$ typically read as in sequences?

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

Flashcard 34: Identify the missing initial condition needed for $f(n)=f(n-1)+f(n-2)$ to start at $n=2$.

Answer: Two starting values, such as $f(0)$ and $f(1)$ (or $f(1)$ and $f(2)$). Second-order recursions need two initial values to start computation.

Flashcard 35: What is the Fibonacci recursion stated in function notation?

Answer: $f(0)=1$, $f(1)=1$, and $f(n+1)=f(n)+f(n-1)$ for $n\ge 1$. Standard Fibonacci definition with two initial conditions and sum rule.

Flashcard 36: What is the domain of an ordinary sequence $a_n$ most commonly taken to be?

Answer: A subset of the integers, usually $\{1,2,3,\dots\}$ or $\{0,1,2,\dots\}$. Standard domains start at 0 or 1 and continue through positive integers.

Flashcard 37: Which option correctly describes why sequences fit the function definition?

Answer: Each integer input $n$ is paired with exactly one output value. Sequences satisfy function definition with unique input-output pairs.

Flashcard 38: What is the value of $f(8)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?

Answer: $f(8)=34$. Apply recursion: $f(8)=f(7)+f(6)=21+13=34$.

Flashcard 39: What is the output $f(3)$ if $f(0)=4$, $f(1)=7$, and $f(n)=f(n-1)+f(n-2)$ for $n\ge 2$?

Answer: $f(3)=18$. Apply recursion: $f(3)=f(2)+f(1)=11+7=18$.

Flashcard 40: What is the value of $f(3)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?

Answer: $f(3)=3$. Apply recursion: $f(3)=f(2)+f(1)=2+1=3$.

Flashcard 41: What is the domain if a sequence is defined only for $-2\le n\le 3$ with integer $n$?

Answer: $\{-2,-1,0,1,2,3\}$. Domain includes all integers within the specified bounds.

Flashcard 42: What is the key difference between a sequence and a continuous function?

Answer: A sequence has integer domain; a continuous function can have real domain. Domain restriction distinguishes discrete sequences from continuous functions.

Flashcard 43: What does the notation $f(n)$ emphasize compared with $a_n$?

Answer: It emphasizes that the sequence is a function of the integer input $n$. Function notation highlights the mapping from input to output.

Flashcard 44: What is the value of $f(5)$ if $f(n)=f(n-1)-2$ with $f(1)=11$?

Answer: $f(5)=3$. Use recursion repeatedly: $f(5)=11-4(2)=11-8=3$.

Flashcard 45: Which set best represents the domain of a sequence defined for $n\ge 0$?

Answer: $\{0,1,2,3,\dots\}$. This set includes all non-negative integers starting from 0.

Flashcard 46: What is the output of the sequence defined by $f(n)=2n+1$ at $n=4$?

Answer: $f(4)=9$. Substitute $n=4$: $f(4)=2(4)+1=9$.

Flashcard 47: What is meant by an initial condition in a recursive sequence?

Answer: A starting value such as $f(0)$ or $f(1)$ needed to begin the recursion. Seeds the recursive process with a known base value.

Flashcard 48: What is the ordered-pair form for the $n$th term of a sequence $f(n)$ on a graph?

Answer: The point $(n,f(n))$. Standard coordinate form with $n$ as input and $f(n)$ as output.

Flashcard 49: What is the output $f(2)$ if $f(0)=4$, $f(1)=7$, and $f(n)=f(n-1)+f(n-2)$ for $n\ge 2$?

Answer: $f(2)=11$. Apply recursion: $f(2)=f(1)+f(0)=7+4=11$.

Flashcard 50: What is the output of the sequence defined by $f(n)=n^2$ at $n=5$?

Answer: $f(5)=25$. Substitute $n=5$: $f(5)=5^2=25$.

Flashcard 51: What is a first-order recursion for a sequence?

Answer: A recursion that defines $f(n)$ using only the previous term, like $f(n)=f(n-1)+c$. First-order means each term depends on exactly one previous term.

Flashcard 52: What is the value of $f(4)$ if $f(n)=2f(n-1)$ with $f(1)=3$?

Answer: $f(4)=24$. Use recursion: $f(2)=2(3)=6$, $f(3)=2(6)=12$, $f(4)=2(12)=24$.

Flashcard 53: What is the value of $f(2)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?

Answer: $f(2)=2$. Apply recursion: $f(2)=f(1)+f(0)=1+1=2$.

Flashcard 54: What is the value of $f(10)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?

Answer: $f(10)=89$. Apply recursion: $f(10)=f(9)+f(8)=55+34=89$.

Flashcard 55: What must be true for a sequence to be a function?

Answer: Each input $n$ in the domain has exactly one output value. Functions require unique outputs for each input in the domain.