Sequences as Functions and Recursion - Algebra 1

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

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

2. What is the range of a sequence?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

14. What is a recursion rule for a sequence?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    Recursive. Each term is defined using the previous term.

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

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

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

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

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

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

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

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

38. What is $f(4)$ for Fibonacci when $f(0)=f(1)=1$ and $f(n+1)=f(n)+f(n-1)$?

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

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

    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.

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

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

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

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

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

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

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

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

44. What is $a_3$ for the explicit sequence $a_n=2^n$ with domain $n\ge 0$?

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

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

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

46. What is the domain of a sequence defined for $n=0,1,2,\dots$?

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

47. What is the key difference between a sequence and a continuous function graph?

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

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

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

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

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

50. What does the notation $a_n$ mean in a sequence?

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

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

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

52. What is an initial condition in a recursive sequence?

    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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