Sequences as Functions and Recursion - Algebra 2
Card 1 of 30
What is the value of $f(3)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
What is the value of $f(3)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
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$f(3)=3$. Apply recursion: $f(3)=f(2)+f(1)=2+1=3$.
$f(3)=3$. Apply recursion: $f(3)=f(2)+f(1)=2+1=3$.
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Identify the correct interpretation of $f(n+1)=f(n)+f(n-1)$ in words.
Identify the correct interpretation of $f(n+1)=f(n)+f(n-1)$ in words.
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Each term equals the sum of the previous two terms. This describes the Fibonacci-type addition pattern in words.
Each term equals the sum of the previous two terms. This describes the Fibonacci-type addition pattern in words.
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What is the ordered-pair form for the $n$-th term of a sequence $f(n)$ on a graph?
What is the ordered-pair form for the $n$-th term of a sequence $f(n)$ on a graph?
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The point $(n,f(n))$. Standard coordinate form with $n$ as input and $f(n)$ as output.
The point $(n,f(n))$. Standard coordinate form with $n$ as input and $f(n)$ as output.
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What is the Fibonacci recursion stated in function notation?
What is the Fibonacci recursion stated in function notation?
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$f(0)=1$, $f(1)=1$, and $f(n+1)=f(n)+f(n-1)$ for $n\ge 1$. Standard Fibonacci definition with two initial conditions and sum rule.
$f(0)=1$, $f(1)=1$, and $f(n+1)=f(n)+f(n-1)$ for $n\ge 1$. Standard Fibonacci definition with two initial conditions and sum rule.
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Which statement correctly describes the range of a sequence as a function?
Which statement correctly describes the range of a sequence as a function?
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The set of all outputs $f(n)$ for integer inputs in the domain. Range consists of all possible function outputs from the domain.
The set of all outputs $f(n)$ for integer inputs in the domain. Range consists of all possible function outputs from the domain.
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Which option correctly states why a recursion alone does not define a unique sequence?
Which option correctly states why a recursion alone does not define a unique sequence?
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Without initial condition(s), many sequences satisfy the same recursion. Initial conditions distinguish between different sequences with same recursion.
Without initial condition(s), many sequences satisfy the same recursion. Initial conditions distinguish between different sequences with same recursion.
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Identify the missing initial condition needed for $f(n)=f(n-1)+f(n-2)$ to start at $n=2$.
Identify the missing initial condition needed for $f(n)=f(n-1)+f(n-2)$ to start at $n=2$.
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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.
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.
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What is the value of $f(5)$ if $f(n)=f(n-1)-2$ with $f(1)=11$?
What is the value of $f(5)$ if $f(n)=f(n-1)-2$ with $f(1)=11$?
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$f(5)=3$. Use recursion repeatedly: $f(5)=11-4(2)=11-8=3$.
$f(5)=3$. Use recursion repeatedly: $f(5)=11-4(2)=11-8=3$.
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What is the value of $f(4)$ if $f(n)=2f(n-1)$ with $f(1)=3$?
What is the value of $f(4)$ if $f(n)=2f(n-1)$ with $f(1)=3$?
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$f(4)=24$. Use recursion: $f(2)=2(3)=6$, $f(3)=2(6)=12$, $f(4)=2(12)=24$.
$f(4)=24$. Use recursion: $f(2)=2(3)=6$, $f(3)=2(6)=12$, $f(4)=2(12)=24$.
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What is the value of $f(3)$ if $f(n)=f(n-1)+4$ with $f(1)=2$?
What is the value of $f(3)$ if $f(n)=f(n-1)+4$ with $f(1)=2$?
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$f(3)=10$. Use recursion: $f(2)=2+4=6$, $f(3)=6+4=10$.
$f(3)=10$. Use recursion: $f(2)=2+4=6$, $f(3)=6+4=10$.
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What is the domain if a sequence is defined only for $-2\le n\le 3$ with integer $n$?
What is the domain if a sequence is defined only for $-2\le n\le 3$ with integer $n$?
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${-2,-1,0,1,2,3}$. Domain includes all integers within the specified bounds.
${-2,-1,0,1,2,3}$. Domain includes all integers within the specified bounds.
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What is the value of $f(3)$ if $f(0)=2$, $f(1)=5$, and $f(n+1)=f(n)+f(n-1)$ for $n\ge 1$?
What is the value of $f(3)$ if $f(0)=2$, $f(1)=5$, and $f(n+1)=f(n)+f(n-1)$ for $n\ge 1$?
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$f(3)=12$. Apply recursion with $n=2$: $f(3)=f(2)+f(1)=7+5=12$.
$f(3)=12$. Apply recursion with $n=2$: $f(3)=f(2)+f(1)=7+5=12$.
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What is the value of $f(5)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
What is the value of $f(5)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
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$f(5)=8$. Apply recursion: $f(5)=f(4)+f(3)=5+3=8$.
$f(5)=8$. Apply recursion: $f(5)=f(4)+f(3)=5+3=8$.
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What is the value of $f(4)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
What is the value of $f(4)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
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$f(4)=5$. Apply recursion: $f(4)=f(3)+f(2)=3+2=5$.
$f(4)=5$. Apply recursion: $f(4)=f(3)+f(2)=3+2=5$.
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What is the value of $f(3)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
What is the value of $f(3)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
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$f(3)=3$. Apply recursion: $f(3)=f(2)+f(1)=2+1=3$.
$f(3)=3$. Apply recursion: $f(3)=f(2)+f(1)=2+1=3$.
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What is the value of $f(2)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
What is the value of $f(2)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
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$f(2)=2$. Apply recursion: $f(2)=f(1)+f(0)=1+1=2$.
$f(2)=2$. Apply recursion: $f(2)=f(1)+f(0)=1+1=2$.
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Identify the correct domain description for the Fibonacci sequence defined by $f(0)=f(1)=1$.
Identify the correct domain description for the Fibonacci sequence defined by $f(0)=f(1)=1$.
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All integers $n$ with $n\ge 0$. Domain starts at 0 since both $f(0)$ and $f(1)$ are defined.
All integers $n$ with $n\ge 0$. Domain starts at 0 since both $f(0)$ and $f(1)$ are defined.
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What must be true for a sequence to be a function?
What must be true for a sequence to be a function?
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Each input $n$ in the domain has exactly one output value. Functions require unique outputs for each input in the domain.
Each input $n$ in the domain has exactly one output value. Functions require unique outputs for each input in the domain.
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What does the notation $f(n)$ emphasize compared with $a_n$?
What does the notation $f(n)$ emphasize compared with $a_n$?
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It emphasizes that the sequence is a function of the integer input $n$. Function notation highlights the mapping from input to output.
It emphasizes that the sequence is a function of the integer input $n$. Function notation highlights the mapping from input to output.
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What is the notation $a_n$ typically read as in sequences?
What is the notation $a_n$ typically read as in sequences?
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The $n$th term of the sequence. Subscript notation emphasizes the term's position in the sequence.
The $n$th term of the sequence. Subscript notation emphasizes the term's position in the sequence.
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Which option correctly describes why sequences fit the function definition?
Which option correctly describes why sequences fit the function definition?
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Each integer input $n$ is paired with exactly one output value. Sequences satisfy function definition with unique input-output pairs.
Each integer input $n$ is paired with exactly one output value. Sequences satisfy function definition with unique input-output pairs.
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What is the correct input set for a sequence term labeled $f(12)$?
What is the correct input set for a sequence term labeled $f(12)$?
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The input is the integer $n=12$. The subscript 12 indicates the integer input to the sequence function.
The input is the integer $n=12$. The subscript 12 indicates the integer input to the sequence function.
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What is the value of $f(7)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
What is the value of $f(7)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
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$f(7)=21$. Apply recursion: $f(7)=f(6)+f(5)=13+8=21$.
$f(7)=21$. Apply recursion: $f(7)=f(6)+f(5)=13+8=21$.
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What is the value of $f(6)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
What is the value of $f(6)$ for Fibonacci given $f(0)=1$, $f(1)=1$, $f(n+1)=f(n)+f(n-1)$?
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$f(6)=13$. Apply recursion: $f(6)=f(5)+f(4)=8+5=13$.
$f(6)=13$. Apply recursion: $f(6)=f(5)+f(4)=8+5=13$.
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What is the meaning of the index $n$ in a sequence written $f(n)$?
What is the meaning of the index $n$ in a sequence written $f(n)$?
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The input that indicates position in the sequence. Index $n$ specifies which term in the sequence ordering.
The input that indicates position in the sequence. Index $n$ specifies which term in the sequence ordering.
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What is the $n$th term notation for the sequence values $f(1),f(2),f(3),\dots$?
What is the $n$th term notation for the sequence values $f(1),f(2),f(3),\dots$?
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The list of outputs indexed by integers: $f(1),f(2),f(3),\dots$. Function notation creates an indexed list of sequence outputs.
The list of outputs indexed by integers: $f(1),f(2),f(3),\dots$. Function notation creates an indexed list of sequence outputs.
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Identify whether $f(n)=\frac{1}{n}$ is defined at $n=0$ when the domain is ${0,1,2,\dots}$.
Identify whether $f(n)=\frac{1}{n}$ is defined at $n=0$ when the domain is ${0,1,2,\dots}$.
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No, because $\frac{1}{0}$ is undefined. Division by zero makes the function undefined at $n=0$.
No, because $\frac{1}{0}$ is undefined. Division by zero makes the function undefined at $n=0$.
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Identify the domain for a sequence defined by $f(n)=\frac{1}{n}$ with the restriction $n\ge 1$.
Identify the domain for a sequence defined by $f(n)=\frac{1}{n}$ with the restriction $n\ge 1$.
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${1,2,3,\dots}$. Domain excludes 0 to avoid division by zero in $\frac{1}{n}$.
${1,2,3,\dots}$. Domain excludes 0 to avoid division by zero in $\frac{1}{n}$.
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What is the output of the sequence defined by $f(n)=\frac{n-1}{2}$ at $n=7$?
What is the output of the sequence defined by $f(n)=\frac{n-1}{2}$ at $n=7$?
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$f(7)=3$. Substitute $n=7$: $f(7)=\frac{7-1}{2}=\frac{6}{2}=3$.
$f(7)=3$. Substitute $n=7$: $f(7)=\frac{7-1}{2}=\frac{6}{2}=3$.
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What is the output of the sequence defined by $f(n)=3\cdot 2^n$ at $n=3$?
What is the output of the sequence defined by $f(n)=3\cdot 2^n$ at $n=3$?
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$f(3)=24$. Substitute $n=3$: $f(3)=3\cdot 2^3=3\cdot 8=24$.
$f(3)=24$. Substitute $n=3$: $f(3)=3\cdot 2^3=3\cdot 8=24$.
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