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 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 2: 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 3: 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 4: 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 5: 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 6: 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 7: 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 8: 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 9: 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 10: 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 11: 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 12: 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 13: 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 14: 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 15: 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 16: 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 17: 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 18: 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.

Flashcard 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: 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 26: 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 27: 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 28: 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 29: 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 30: What is the output of the sequence defined by $f(n)=3\cdot 2^n$ at $n=3$?

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