Sequences as Functions and Recursion - Algebra
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What is $f(4)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?
What is $f(4)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?
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$f(4)=5$. Using $f(3)=3$ and $f(2)=2$: $f(4)=f(3)+f(2)=3+2=5$.
$f(4)=5$. Using $f(3)=3$ and $f(2)=2$: $f(4)=f(3)+f(2)=3+2=5$.
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What does it mean for a sequence to be defined explicitly?
What does it mean for a sequence to be defined explicitly?
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A formula gives $a_n$ directly in terms of $n$. No need for previous terms; just substitute $n$ into the formula.
A formula gives $a_n$ directly in terms of $n$. No need for previous terms; just substitute $n$ into the formula.
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What is the range of a sequence?
What is the range of a sequence?
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The set of term values ${f(n)}$ produced by allowed integer $n$. The range contains all possible output values of the function.
The set of term values ${f(n)}$ produced by allowed integer $n$. The range contains all possible output values of the function.
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What is the key difference between a sequence and a continuous function graph?
What is the key difference between a sequence and a continuous function graph?
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A sequence has discrete integer inputs, not all real $x$-values. Sequences only exist at integer points, not between them.
A sequence has discrete integer inputs, not all real $x$-values. Sequences only exist at integer points, not between them.
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What does the notation $a_n$ mean in a sequence?
What does the notation $a_n$ mean in a sequence?
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The $n$th term of the sequence. Subscript notation indicates the term's position in the sequence.
The $n$th term of the sequence. Subscript notation indicates the term's position in the sequence.
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What does the notation $f(n)$ represent when $f$ is a sequence?
What does the notation $f(n)$ represent when $f$ is a sequence?
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The output (term) of the sequence at integer input $n$. The function notation shows the sequence value at position $n$.
The output (term) of the sequence at integer input $n$. The function notation shows the sequence value at position $n$.
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What is the domain of a sequence defined for $n=0,1,2,\dots$?
What is the domain of a sequence defined for $n=0,1,2,\dots$?
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The nonnegative integers ${0,1,2,\dots}$. Starting at $n=0$ includes zero in the domain.
The nonnegative integers ${0,1,2,\dots}$. Starting at $n=0$ includes zero in the domain.
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What is the domain of a typical sequence written as $a_n$ for $n=1,2,3,\dots$?
What is the domain of a typical sequence written as $a_n$ for $n=1,2,3,\dots$?
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The positive integers ${1,2,3,\dots}$. Starting at $n=1$ gives positive integers as the domain.
The positive integers ${1,2,3,\dots}$. Starting at $n=1$ gives positive integers as the domain.
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What is the definition of a sequence as a function in Algebra 1?
What is the definition of a sequence as a function in Algebra 1?
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A function with domain a subset of the integers, usually ${0,1,2,\dots}$. The domain must be integers, not continuous real numbers.
A function with domain a subset of the integers, usually ${0,1,2,\dots}$. The domain must be integers, not continuous real numbers.
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Identify whether $a_1=1$ and $a_n=a_{n-1}+3$ (for $n\ge 2$) is explicit or recursive.
Identify whether $a_1=1$ and $a_n=a_{n-1}+3$ (for $n\ge 2$) is explicit or recursive.
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Recursive. Each term is defined using the previous term.
Recursive. Each term is defined using the previous term.
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Identify whether $a_n=2n-1$ (for $n\ge 1$) is explicit or recursive.
Identify whether $a_n=2n-1$ (for $n\ge 1$) is explicit or recursive.
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Explicit. The formula directly gives $a_n$ in terms of $n$.
Explicit. The formula directly gives $a_n$ in terms of $n$.
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Find $a_3$ if $a_0=2$, $a_1=3$, and $a_n=a_{n-1}+a_{n-2}$ for $n\ge 2$.
Find $a_3$ if $a_0=2$, $a_1=3$, and $a_n=a_{n-1}+a_{n-2}$ for $n\ge 2$.
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$a_3=8$. Adding the two previous terms: $a_3=a_2+a_1=5+3=8$.
$a_3=8$. Adding the two previous terms: $a_3=a_2+a_1=5+3=8$.
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What is required to define a recursive sequence that uses $a_{n-1}$ only?
What is required to define a recursive sequence that uses $a_{n-1}$ only?
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One initial condition (for example, $a_1$) and the recursion rule. One previous term requires one starting value to begin.
One initial condition (for example, $a_1$) and the recursion rule. One previous term requires one starting value to begin.
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Find $a_2$ if $a_0=2$, $a_1=3$, and $a_n=a_{n-1}+a_{n-2}$ for $n\ge 2$.
Find $a_2$ if $a_0=2$, $a_1=3$, and $a_n=a_{n-1}+a_{n-2}$ for $n\ge 2$.
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$a_2=5$. Adding the two previous terms: $a_2=a_1+a_0=3+2=5$.
$a_2=5$. Adding the two previous terms: $a_2=a_1+a_0=3+2=5$.
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Find $a_3$ if $a_0=1$ and $a_n=3a_{n-1}$ for $n\ge 1$.
Find $a_3$ if $a_0=1$ and $a_n=3a_{n-1}$ for $n\ge 1$.
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$a_3=27$. $a_1=3(1)=3$, $a_2=3(3)=9$, $a_3=3(9)=27$.
$a_3=27$. $a_1=3(1)=3$, $a_2=3(3)=9$, $a_3=3(9)=27$.
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Find $a_2$ if $a_0=5$ and $a_n=2a_{n-1}$ for $n\ge 1$.
Find $a_2$ if $a_0=5$ and $a_n=2a_{n-1}$ for $n\ge 1$.
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$a_2=20$. $a_1=2(5)=10$, $a_2=2(10)=20$.
$a_2=20$. $a_1=2(5)=10$, $a_2=2(10)=20$.
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Find $a_3$ if $a_1=10$ and $a_n=a_{n-1}-4$ for $n\ge 2$.
Find $a_3$ if $a_1=10$ and $a_n=a_{n-1}-4$ for $n\ge 2$.
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$a_3=2$. $a_2=10-4=6$, $a_3=6-4=2$.
$a_3=2$. $a_2=10-4=6$, $a_3=6-4=2$.
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Find $a_4$ if $a_1=3$ and $a_n=a_{n-1}+2$ for $n\ge 2$.
Find $a_4$ if $a_1=3$ and $a_n=a_{n-1}+2$ for $n\ge 2$.
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$a_4=9$. $a_2=3+2=5$, $a_3=5+2=7$, $a_4=7+2=9$.
$a_4=9$. $a_2=3+2=5$, $a_3=5+2=7$, $a_4=7+2=9$.
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What is required to define a recursive sequence that uses $a_{n-1}$ and $a_{n-2}$?
What is required to define a recursive sequence that uses $a_{n-1}$ and $a_{n-2}$?
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Two initial conditions (for example, $a_0$ and $a_1$) and the rule. Two previous terms require two starting values to begin.
Two initial conditions (for example, $a_0$ and $a_1$) and the rule. Two previous terms require two starting values to begin.
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What is the input variable for a sequence written as $a_n$?
What is the input variable for a sequence written as $a_n$?
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The index $n$, which is an integer in the domain. The subscript $n$ represents the independent variable (input).
The index $n$, which is an integer in the domain. The subscript $n$ represents the independent variable (input).
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What is the ordered-pair representation of a sequence term $a_n$ on a graph?
What is the ordered-pair representation of a sequence term $a_n$ on a graph?
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The point $(n,a_n)$. The $x$-coordinate is $n$, the $y$-coordinate is the term value.
The point $(n,a_n)$. The $x$-coordinate is $n$, the $y$-coordinate is the term value.
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Which representation best matches a sequence: discrete points or a continuous curve?
Which representation best matches a sequence: discrete points or a continuous curve?
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Discrete points $(n,a_n)$ at integer $n$. Sequences have gaps between integer inputs, unlike continuous functions.
Discrete points $(n,a_n)$ at integer $n$. Sequences have gaps between integer inputs, unlike continuous functions.
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What does the condition $n\ge 1$ mean in a recursion like $f(n+1)=f(n)+f(n-1)$?
What does the condition $n\ge 1$ mean in a recursion like $f(n+1)=f(n)+f(n-1)$?
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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.
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.
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Identify the domain of Fibonacci as defined by $f(0)$, $f(1)$, and $f(n+1)$ for $n\ge 1$.
Identify the domain of Fibonacci as defined by $f(0)$, $f(1)$, and $f(n+1)$ for $n\ge 1$.
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All integers $n\ge 0$. Initial conditions define $f(0)$ and $f(1)$, rule applies for $n\ge 1$.
All integers $n\ge 0$. Initial conditions define $f(0)$ and $f(1)$, rule applies for $n\ge 1$.
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What is $f(5)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?
What is $f(5)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?
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$f(5)=8$. Using $f(4)=5$ and $f(3)=3$: $f(5)=f(4)+f(3)=5+3=8$.
$f(5)=8$. Using $f(4)=5$ and $f(3)=3$: $f(5)=f(4)+f(3)=5+3=8$.
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What does it mean for a sequence to be defined recursively?
What does it mean for a sequence to be defined recursively?
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Terms are defined using previous term(s) plus initial condition(s). Each term depends on one or more preceding terms.
Terms are defined using previous term(s) plus initial condition(s). Each term depends on one or more preceding terms.
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What is the recursive definition of the Fibonacci sequence given in the standard?
What is the recursive definition of the Fibonacci sequence given in the standard?
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$f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$ for $n\ge 1$. This is the standard recursive definition from CCSS.F-IF.3.
$f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$ for $n\ge 1$. This is the standard recursive definition from CCSS.F-IF.3.
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What is $f(3)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?
What is $f(3)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?
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$f(3)=3$. Using $f(2)=2$ and $f(1)=1$: $f(3)=f(2)+f(1)=2+1=3$.
$f(3)=3$. Using $f(2)=2$ and $f(1)=1$: $f(3)=f(2)+f(1)=2+1=3$.
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What is $f(2)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?
What is $f(2)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?
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$f(2)=2$. Using $f(1)=1$ and $f(0)=1$: $f(2)=f(1)+f(0)=1+1=2$.
$f(2)=2$. Using $f(1)=1$ and $f(0)=1$: $f(2)=f(1)+f(0)=1+1=2$.
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What is an initial condition in a recursive sequence?
What is an initial condition in a recursive sequence?
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Starting value(s) like $a_1$ or $a_0$ needed to generate later terms. Without initial values, the recursive pattern cannot begin.
Starting value(s) like $a_1$ or $a_0$ needed to generate later terms. Without initial values, the recursive pattern cannot begin.
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