Sequences as Functions and Recursion - Algebra 2

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

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

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

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

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

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

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 $n\ge 1$?

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

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

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

6. What is an explicit formula for a sequence?

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

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

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

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

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

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

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

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

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

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 $n\ge 1$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

19. What is a second-order recursion for a sequence?

    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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

28. What is a recursive definition of a sequence?

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

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

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

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

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

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

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

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

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

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

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

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

    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.

35. What is the Fibonacci recursion stated in function notation?

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

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

    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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

51. What is a first-order recursion for a sequence?

    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.

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

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

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

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

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

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

55. What must be true for a sequence to be a function?

    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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