Algebra 2 Quiz: Sequences As Functions And Recursion
20 questions · exam conditions
0:00
Sequences As Functions And RecursionQuestion 1 of 20

Consider the recursively defined sequence b0=5b_0 = 5, b1=2b_1 = 2, and bn=bn−1+bn−2b_n = b_{n-1} + b_{n-2} for n≥2n \geq 2. Which statement about this sequence as a function is most accurate?

The function requires exactly one previous term to determine each subsequent value
The function requires exactly two previous terms to determine each subsequent value
The function can be expressed as f(n)=5n+2f(n) = 5n + 2 for all valid inputs
The function has an undefined value at n=0n = 0 and n=1n = 1
← Back to quizzes

Algebra 2 Quiz

Algebra 2 Quiz: Sequences As Functions And Recursion

Practice Sequences As Functions And Recursion in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Sequences As Functions And Recursion, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Consider the recursively defined sequence b0=5b_0 = 5, b1=2b_1 = 2, and bn=bn−1+bn−2b_n = b_{n-1} + b_{n-2} for n≥2n \geq 2. Which statement about this sequence as a function is most accurate?

  1. The function requires exactly one previous term to determine each subsequent value
  2. The function requires exactly two previous terms to determine each subsequent value (correct answer)
  3. The function can be expressed as f(n)=5n+2f(n) = 5n + 2 for all valid inputs
  4. The function has an undefined value at n=0n = 0 and n=1n = 1
Explanation: This is a Fibonacci-type sequence where each term depends on the sum of the two preceding terms, so exactly two previous terms are needed for n≥2. Choice A is wrong because one term isn't sufficient. Choice C is wrong because this recursive relationship doesn't produce a linear function. Choice D is wrong because b₀=5 and b₁=2 are explicitly defined initial conditions.

Question 2

A sequence is a function whose domain is a subset of the integers, so it is defined only at discrete inputs. Which set is the most appropriate domain for a sequence written as a1,a2,a3,…a_1, a_2, a_3, \dots?

  1. All rational numbers Q\mathbb{Q}
  2. All integers Z\mathbb{Z}
  3. All real numbers R\mathbb{R}
  4. All positive integers {1,2,3,… }\{1,2,3,\dots\} (correct answer)
Explanation: This question tests your understanding that sequences are special functions where the domain is a subset of the integers—meaning they're only defined for whole number inputs like 1, 2, 3, not for values in between. A sequence is a function whose inputs are integers (often starting at 1 or 0) and whose outputs are the sequence terms: f(1) = first term, f(2) = second term, etc. Unlike continuous functions that have values for all real numbers, sequences have values only at integer positions. If you graphed a sequence, you'd see separate dots (like (1, a₁), (2, a₂), (3, a₃)), not a connected curve. This discrete nature makes sequences fundamentally different from functions like f(x) = x²! For a sequence like a₁, a₂, a₃, ..., the most appropriate domain is the positive integers {1,2,3,...}, as it matches the indexing starting from 1. Choice C correctly defines the integer domain as positive integers. A distractor like A (all reals) fails because sequences are discrete, not defined for non-integers—great job spotting that! Remember, sequences aren't continuous; their domain restricts to integers for that dotted graph effect. Keep exploring, and these concepts will click!

Question 3

The sequence 1,3,9,27,81,…1,3,9,27,81,\dots is a function with domain n∈{1,2,3,… }n\in\{1,2,3,\dots\}. Which recursive definition matches this sequence?

  1. a1=3, an+1=3ana_1=3,\ a_{n+1}=3a_n
  2. a1=1, an+1=an+3a_1=1,\ a_{n+1}=a_n+3
  3. a1=1, an+1=3ana_1=1,\ a_{n+1}=3a_n (correct answer)
  4. a1=1, an+1=an3a_1=1,\ a_{n+1}=a_n^3
Explanation: This question tests your understanding that sequences are special functions where the domain is a subset of the integers—meaning they're only defined for whole number inputs like 1, 2, 3, not for values in between. A sequence is a function whose inputs are integers (often starting at 1 or 0) and whose outputs are the sequence terms: a₁ = first term, a₂ = second term, etc. This geometric sequence 1, 3, 9, 27, 81,... multiplies by 3 each time, so the recursive rule should reflect that constant ratio. Checking each option: For B, a₁ = 1, a₂ = 31 = 3, a₃ = 33 = 9, a₄ = 27, a₅ = 81, matching perfectly. Choice B correctly defines with initial value and rule as a₁=1, a_{n+1}=3a_n. Option A fails because it adds 3 each time, giving 1,4,7,10,... which is arithmetic, not geometric—remember to distinguish between adding a constant versus multiplying by one. Recursive vs explicit: recursive definitions are often simpler to state ('multiply by 3') but require step-by-step for later terms, while an explicit formula like a_n = 3^{n-1} lets you compute directly, like a_5 = 3^4 = 81 instantly.

Question 4

A sequence can be viewed as a discrete function ff whose domain is the positive integers n∈{1,2,3,… }n \in \{1,2,3,\dots\}. For the Fibonacci sequence defined by f(1)=f(2)=1f(1)=f(2)=1 and f(n)=f(n−1)+f(n−2)f(n)=f(n-1)+f(n-2) for n≥3n\ge 3, what is f(7)f(7)?​

  1. 1111
  2. 2121
  3. 88
  4. 1313 (correct answer)
Explanation: This question tests your understanding that sequences are special functions where the domain is a subset of the integers—meaning they're only defined for whole number inputs like 1, 2, 3, not for values in between. A sequence is a function whose inputs are integers (often starting at 1 or 0) and whose outputs are the sequence terms: f(1) = first term, f(2) = second term, etc. Unlike continuous functions that have values for all real numbers, sequences have values only at integer positions. If you graphed a sequence, you'd see separate dots (like (1, a₁), (2, a₂), (3, a₃)), not a connected curve. This discrete nature makes sequences fundamentally different from functions like f(x) = x²! Recursive definitions specify how to get each term from previous term(s): the Fibonacci sequence is the classic example with f(1) = 1, f(2) = 1 and f(n) = f(n-1) + f(n-2) for n ≥ 3, meaning each term equals the sum of the two before it. To find f(7), you can't jump there directly—you must calculate f(3) = f(2) + f(1) = 1 + 1 = 2, then f(4) = 2 + 1 = 3, then f(5) = 3 + 2 = 5, f(6) = 5 + 3 = 8, finally f(7) = 8 + 5 = 13. Choice C correctly evaluates recursively as 13. A common distractor like D (21) might come from miscounting terms or starting from f(0), but remember to use the given initial values precisely. Evaluating recursive sequences systematically: (1) Write down the initial value(s) clearly: f(1) = 1, f(2) = 1, (2) Set up a table or list: n = 1, 2, 3, ... down one side, (3) Apply the recursive rule one step at a time: for Fibonacci, f(3) = f(2) + f(1), so look up f(2) and f(1) from what you've already calculated, add them, write the result, (4) Continue until you reach the desired term. Don't skip steps—recursion is sequential by nature! Keep practicing, and you'll master these quickly!

Question 5

A sequence can be viewed as a function whose domain is a subset of the integers. For the Fibonacci sequence defined on n∈{1,2,3,… }n \in \{1,2,3,\dots\} by f(1)=1f(1)=1, f(2)=1f(2)=1, and f(n)=f(n−1)+f(n−2)f(n)=f(n-1)+f(n-2) for n≥3n\ge 3, find f(8)f(8).

  1. 1313
  2. 3434
  3. 2121 (correct answer)
  4. 88
Explanation: This question tests your understanding that sequences are special functions where the domain is a subset of the integers—meaning they're only defined for whole number inputs like 1, 2, 3, not for values in between. Recursive definitions specify how to get each term from previous term(s): the Fibonacci sequence is the classic example with f(1)=1f(1) = 1, f(2)=1f(2) = 1 and f(n)=f(n−1)+f(n−2)f(n) = f(n-1) + f(n-2), meaning each term equals the sum of the two before it. To find f(8)f(8), start with f(1)=1f(1)=1, f(2)=1f(2)=1, then f(3)=2f(3)=2, f(4)=3f(4)=3, f(5)=5f(5)=5, f(6)=8f(6)=8, f(7)=13f(7)=13, and f(8)=21f(8)=21—building step by step ensures accuracy. Choice C correctly evaluates recursively as 21. A common distractor like choice A (13) might come from stopping at f(7)f(7) instead of f(8)f(8), so always verify the term number. Evaluating recursive sequences systematically: (1) Write down the initial value(s) clearly: f(1)=1f(1)=1, f(2)=1f(2)=1, (2) Set up a table or list: n=1,2,3,..., (3) Apply the recursive rule one step at a time: for Fibonacci, add the previous two, (4) Continue until you reach the desired term—don't skip steps, recursion is sequential by nature! Recursive vs explicit: recursive definitions are often simpler to state but require calculating all prior terms, while explicit formulas allow direct computation, though for Fibonacci, the explicit form (Binet's) is more complex—keep practicing both!

Question 6

Two students define sequences differently. Student A: xn=2n+1x_n = 2n + 1 for n≥1n \geq 1. Student B: y1=3y_1 = 3 and yn=yn−1+2y_n = y_{n-1} + 2 for n≥2n \geq 2. Which statement about these sequences as functions is most accurate?

  1. Only Student A's sequence represents a function since it has an explicit formula
  2. Student B's sequence is not a function because it's defined recursively rather than explicitly
  3. Both sequences represent the same function with domain of positive integers (correct answer)
  4. The sequences represent different functions because they use different definition methods
Explanation: Both sequences generate the same values: 3, 5, 7, 9, 11, ... Student A uses explicit formula x_n = 2n+1. Student B uses recursion but y_n = 3+2(n-1) = 2n+1. They represent the same function. Choice A is wrong because recursive definitions also represent functions. Choice B is wrong because recursively defined sequences are functions. Choice D is wrong because the same function can have multiple representations.

Question 7

A population model is defined by P0=100P_0 = 100 and Pn=1.2Pn−1−5P_n = 1.2P_{n-1} - 5 for n≥1n \geq 1. If we interpret this as a function g(n)=Png(n) = P_n, what is g(3)−g(1)g(3) - g(1)?

  1. 31.631.6 (correct answer)
  2. 33.633.6
  3. 36.836.8
  4. 38.838.8
Explanation: First calculate the terms: P₁ = 1.2(100) - 5 = 115, P₂ = 1.2(115) - 5 = 133, P₃ = 1.2(133) - 5 = 154.6. So g(3) - g(1) = 154.6 - 115 = 31.6. Choice B results from calculation errors in the recursive steps. Choice C comes from incorrectly computing P₃ - P₀. Choice D results from sign errors in the recursive formula.

Question 8

Two sequences are defined as follows: Sequence A: u1=4u_1 = 4, un=un−1+3u_n = u_{n-1} + 3 for n≥2n \geq 2. Sequence B: v1=4v_1 = 4, vn=3vn−1v_n = 3v_{n-1} for n≥2n \geq 2. When comparing these sequences as functions, which statement is true?

  1. Both sequences can be written as explicit linear functions of nn
  2. Sequence A grows exponentially while Sequence B grows linearly
  3. Sequence A can be written as u(n)=3n+1u(n) = 3n + 1 and Sequence B grows exponentially (correct answer)
  4. Both sequences have the same rate of growth for large values of nn
Explanation: Sequence A is arithmetic: u(n) = 4 + 3(n-1) = 3n + 1, which is linear. Sequence B is geometric: v(n) = 4·3^(n-1), which is exponential. Choice A is wrong because B is exponential, not linear. Choice B reverses the growth types. Choice D is wrong because exponential growth eventually dominates linear growth.

Question 9

Consider the sequence t1=8t_1 = 8, t2=12t_2 = 12, and tn=tn−1+tn−22t_n = \frac{t_{n-1} + t_{n-2}}{2} for n≥3n \geq 3. If this represents a function h(n)=tnh(n) = t_n, which property best describes the long-term behavior?

  1. The function values increase without bound as nn approaches infinity
  2. The function values approach a finite limit as nn approaches infinity (correct answer)
  3. The function values oscillate between 8 and 12 for all n≥3n \geq 3
  4. The function values decrease without bound as nn approaches infinity
Explanation: Since each term is the average of the two preceding terms, the sequence converges to the average of the initial terms: (8+12)/2 = 10. This is a convergent sequence. Choice A is wrong because averaging doesn't create unbounded growth. Choice C is wrong because the values don't oscillate but converge. Choice D is wrong because the terms approach 10, not negative infinity.

Question 10

A sequence is defined recursively where c1=5c_1 = 5 and cn=(−1)n⋅cn−1+2c_n = (-1)^n \cdot c_{n-1} + 2 for n≥2n \geq 2. If we consider the function f(n)=cnf(n) = c_n, what is the relationship between f(4)f(4) and f(6)f(6)?

  1. f(4)=f(6)−4f(4) = f(6) - 4
  2. f(4)=−f(6)f(4) = -f(6)
  3. f(4)=f(6)+4f(4) = f(6) + 4
  4. f(4)=f(6)f(4) = f(6) (correct answer)
Explanation: When you encounter a recursive sequence, the key is to calculate the first several terms systematically to identify any patterns. This sequence uses both the previous term and a sign that alternates based on whether nn is even or odd. Let's calculate the terms step by step using cn=(−1)n⋅cn−1+2c_n = (-1)^n \cdot c_{n-1} + 2: c1=5c_1 = 5 (given) c2=(−1)2⋅c1+2=1⋅5+2=7c_2 = (-1)^2 \cdot c_1 + 2 = 1 \cdot 5 + 2 = 7 c3=(−1)3⋅c2+2=−1⋅7+2=−5c_3 = (-1)^3 \cdot c_2 + 2 = -1 \cdot 7 + 2 = -5 c4=(−1)4⋅c3+2=1⋅(−5)+2=−3c_4 = (-1)^4 \cdot c_3 + 2 = 1 \cdot (-5) + 2 = -3 c5=(−1)5⋅c4+2=−1⋅(−3)+2=5c_5 = (-1)^5 \cdot c_4 + 2 = -1 \cdot (-3) + 2 = 5 c6=(−1)6⋅c5+2=1⋅5+2=7c_6 = (-1)^6 \cdot c_5 + 2 = 1 \cdot 5 + 2 = 7 Notice that c5=c1=5c_5 = c_1 = 5 and c6=c2=7c_6 = c_2 = 7. The sequence repeats every 4 terms! Therefore f(4)=c4=−3f(4) = c_4 = -3 and f(6)=c6=−3f(6) = c_6 = -3, making choice D correct. Choice A suggests f(4)=f(6)−4f(4) = f(6) - 4, which would mean −3=−3−4=−7-3 = -3 - 4 = -7, which is false. Choice B claims f(4)=−f(6)f(4) = -f(6), meaning −3=−(−3)=3-3 = -(-3) = 3, also incorrect. Choice C proposes f(4)=f(6)+4f(4) = f(6) + 4, giving us −3=−3+4=1-3 = -3 + 4 = 1, which doesn't work either. Study tip: With recursive sequences involving alternating signs, always calculate enough terms to spot the pattern. Many recursive sequences are periodic, so look for when values start repeating.

Question 11

A sequence can be viewed as a discrete function whose domain is the positive integers. For the Fibonacci sequence defined by f(1)=f(2)=1f(1)=f(2)=1 and f(n)=f(n−1)+f(n−2)f(n)=f(n-1)+f(n-2) for n≥3n\ge 3, find f(7)f(7).

  1. 2121
  2. 88
  3. 1313 (correct answer)
  4. 1111
Explanation: This question tests your understanding that sequences are special functions where the domain is a subset of the integers—meaning they're only defined for whole number inputs like 1, 2, 3, not for values in between. A sequence is a function whose inputs are integers (often starting at 1 or 0) and whose outputs are the sequence terms: f(1) = first term, f(2) = second term, etc. Unlike continuous functions that have values for all real numbers, sequences have values only at integer positions. If you graphed a sequence, you'd see separate dots (like (1, a₁), (2, a₂), (3, a₃)), not a connected curve. This discrete nature makes sequences fundamentally different from functions like f(x) = x²! Recursive definitions specify how to get each term from previous term(s): the Fibonacci sequence is the classic example with f(1) = 1, f(2) = 1 and f(n) = f(n-1) + f(n-2) for n ≥ 3, meaning each term equals the sum of the two before it. To find f(7), you can't jump there directly—you must calculate f(3) = f(2) + f(1) = 1 + 1 = 2, then f(4) = 2 + 1 = 3, f(5) = 3 + 2 = 5, f(6) = 5 + 3 = 8, finally f(7) = 8 + 5 = 13. Choice C correctly evaluates recursively as 13. A common distractor like D (21) might come from mistakenly adding more terms or skipping steps, but remember to build sequentially from the initials. Evaluating recursive sequences systematically: (1) Write down the initial value(s) clearly: f(1) = 1, f(2) = 1, (2) Set up a table or list: n = 1, 2, 3, ... down one side, (3) Apply the recursive rule one step at a time: for Fibonacci, f(3) = f(2) + f(1), so look up f(2) and f(1) from what you've already calculated, add them, write the result, (4) Continue until you reach the desired term. Don't skip steps—recursion is sequential by nature! Recursive vs explicit: recursive definitions are often simpler to state ('add the previous two terms') but harder to evaluate for large n (must calculate all terms before it). Explicit formulas are harder to find ('aₙ = formula just in n') but let you jump to any term: a₁₀₀ directly without finding a₁ through a₉₉. For Fibonacci, there IS an explicit formula (Binet's formula involving golden ratio), but it's complex—recursion is actually clearer! Each form has trade-offs.

Question 12

A sequence can be viewed as a discrete function whose domain is a subset of the integers. For the Fibonacci sequence defined by f(1)=f(2)=1f(1)=f(2)=1 and f(n)=f(n−1)+f(n−2)f(n)=f(n-1)+f(n-2) for n≥3n\ge 3 (with domain n∈{1,2,3,… }n\in\{1,2,3,\dots\}), find f(7)f(7).

  1. f(7)=8f(7)=8
  2. f(7)=21f(7)=21
  3. f(7)=13f(7)=13 (correct answer)
  4. f(7)=12f(7)=12
Explanation: This question tests your understanding that sequences are special functions where the domain is a subset of the integers—meaning they're only defined for whole number inputs like 1, 2, 3, not for values in between. Recursive definitions specify how to get each term from previous term(s): the Fibonacci sequence is the classic example with f(1) = f(2) = 1 and f(n) = f(n-1) + f(n-2) for n ≥ 3, meaning each term equals the sum of the two before it. To find f(7), you can't jump there directly—you must calculate f(3) = f(2) + f(1) = 2, then f(4) = 3, f(5) = 5, f(6) = 8, f(7) = 13. Recursion requires building up step by step from the initial values! Let's evaluate step by step: f(1)=1, f(2)=1, f(3)=1+1=2, f(4)=2+1=3, f(5)=3+2=5, f(6)=5+3=8, f(7)=8+5=13. Choice C correctly evaluates recursively as f(7)=13. A common mistake might be stopping one step early, like choice A=8 which is f(6), but keep counting carefully—you're doing great! Evaluating recursive sequences systematically: (1) Write down the initial value(s) clearly: f(1)=1, f(2)=1, (2) Set up a table or list: n=1,2,3,..., (3) Apply the recursive rule one step at a time: for Fibonacci, f(3)=f(2)+f(1), so look up f(2) and f(1) from what you've already calculated, add them, write the result, (4) Continue until you reach the desired term. Don't skip steps—recursion is sequential by nature!

Question 13

A sequence can be viewed as a discrete function ff whose domain is the positive integers n∈{1,2,3,… }n \in \{1,2,3,\dots\}. For the Fibonacci sequence defined by f(1)=f(2)=1f(1)=f(2)=1 and f(n)=f(n−1)+f(n−2)f(n)=f(n-1)+f(n-2) for n≥3n\ge 3, what is f(7)f(7)?

  1. 2121
  2. 88
  3. 1111
  4. 1313 (correct answer)
Explanation: This question tests your understanding that sequences are special functions where the domain is a subset of the integers—meaning they're only defined for whole number inputs like 1, 2, 3, not for values in between. A sequence is a function whose inputs are integers (often starting at 1 or 0) and whose outputs are the sequence terms: f(1)f(1) = first term, f(2)f(2) = second term, etc. Unlike continuous functions that have values for all real numbers, sequences have values only at integer positions. If you graphed a sequence, you'd see separate dots (like (11, a1a_1), (22, a2a_2), (33, a3a_3)), not a connected curve. This discrete nature makes sequences fundamentally different from functions like f(x)=x2f(x) = x^2! Recursive definitions specify how to get each term from previous term(s): the Fibonacci sequence is the classic example with f(1)=1f(1) = 1, f(2)=1f(2) = 1 and f(n)=f(n−1)+f(n−2)f(n) = f(n-1) + f(n-2) for n≥3n \geq 3, meaning each term equals the sum of the two before it. To find f(7)f(7), you can't jump there directly—you must calculate f(3)=f(2)+f(1)=1+1=2f(3) = f(2) + f(1) = 1 + 1 = 2, then f(4)=2+1=3f(4) = 2 + 1 = 3, then f(5)=3+2=5f(5) = 3 + 2 = 5, f(6)=5+3=8f(6) = 5 + 3 = 8, finally f(7)=8+5=13f(7) = 8 + 5 = 13. Choice C correctly evaluates recursively as 13. A common distractor like D (21) might come from miscounting terms or starting from f(0)f(0), but remember to use the given initial values precisely. Evaluating recursive sequences systematically: (1) Write down the initial value(s) clearly: f(1)=1f(1) = 1, f(2)=1f(2) = 1, (2) Set up a table or list: n=1,2,3,…n = 1, 2, 3, \dots down one side, (3) Apply the recursive rule one step at a time: for Fibonacci, f(3)=f(2)+f(1)f(3) = f(2) + f(1), so look up f(2)f(2) and f(1)f(1) from what you've already calculated, add them, write the result, (4) Continue until you reach the desired term. Don't skip steps—recursion is sequential by nature! Keep practicing, and you'll master these quickly!

Question 14

A sequence is defined by s2=7s_2 = 7, s3=11s_3 = 11, and sn=sn−1+sn−2s_n = s_{n-1} + s_{n-2} for n≥4n \geq 4. As a function f(n)=snf(n) = s_n, what is the most restrictive description of the domain?

  1. The set {2,3,4,5,6,...}\{2, 3, 4, 5, 6, ...\} (correct answer)
  2. The set {0,1,2,3,4,5,...}\{0, 1, 2, 3, 4, 5, ...\}
  3. The set {4,5,6,7,8,...}\{4, 5, 6, 7, 8, ...\}
  4. The set {1,2,3,4,5,6,...}\{1, 2, 3, 4, 5, 6, ...\}
Explanation: The sequence is defined starting at n=2 and n=3 as initial conditions, with the recursive rule applying for n≥4. Therefore, the function is defined for all integers n≥2. Choice B incorrectly includes undefined values at n=0 and n=1. Choice C incorrectly excludes the given initial conditions at n=2 and n=3. Choice D incorrectly includes n=1 where the sequence is undefined.

Question 15

A sequence is a function defined only at integer inputs. Which set could be the domain of a sequence a(n)a(n) (viewed as a discrete function)?

  1. {x∈R:x≥0}\{x\in\mathbb{R}: x\ge 0\}
  2. {1,2,3,4,… }\{1,2,3,4,\dots\} (correct answer)
  3. All real numbers between 0 and 1, inclusive
  4. All real numbers R\mathbb{R}
Explanation: This question tests your understanding that sequences are special functions where the domain is a subset of the integers—meaning they're only defined for whole number inputs like 1, 2, 3, not for values in between. A sequence is a function whose inputs are integers (often starting at 1 or 0) and whose outputs are the sequence terms: a(1) = first term, a(2) = second term, etc. Unlike continuous functions that have values for all real numbers, sequences have values only at integer positions. If you graphed a sequence, you'd see separate dots (like (1, a1), (2, a2), (3, a3)), not a connected curve. This discrete nature makes sequences fundamentally different from functions like f(x) = x²! The set {1,2,3,4,...} is a perfect domain for a sequence since it's a subset of positive integers, allowing discrete evaluation like a(1), a(2), etc. Choice B correctly explains the integer domain. Choice A fails because all real numbers would imply a continuous function, not discrete—keep up the good work distinguishing them! Remember, sequences emphasize the ordered list aspect with integer indices.

Question 16

The sequence 1,3,9,27,81,…1,3,9,27,81,\dots can be treated as a function g(n)g(n) with domain n∈{1,2,3,… }n\in\{1,2,3,\dots\}. Which recursive definition correctly generates this sequence?

  1. g0=1,  gn+1=gn+gn−1g_0=1,\; g_{n+1}=g_n+g_{n-1}
  2. g1=1,  gn+1=gn+3g_1=1,\; g_{n+1}=g_n+3
  3. g1=1,  gn+1=3gng_1=1,\; g_{n+1}=3g_n (correct answer)
  4. gn+1=3gng_{n+1}=3g_n
Explanation: This question tests your understanding that sequences are special functions where the domain is a subset of the integers—meaning they're only defined for whole number inputs like 1, 2, 3, not for values in between. A sequence is a function whose inputs are integers (often starting at 1 or 0) and whose outputs are the sequence terms: g(1) = first term, g(2) = second term, etc. Unlike continuous functions that have values for all real numbers, sequences have values only at integer positions. If you graphed a sequence, you'd see separate dots (like (1,1), (2,3), (3,9)), not a connected curve. This discrete nature makes sequences fundamentally different from functions like f(x) = x²! For this geometric sequence, choice B generates it perfectly: g(1)=1, g(2)=31=3, g(3)=33=9, g(4)=39=27, g(5)=327=81, matching exactly. Choice B correctly defines with initial value and rule as g(n+1)=3g(n). Choice A fails because it produces an arithmetic sequence: 1,4,7,10,..., not matching the given terms—great job spotting the difference! Recursive vs explicit: recursive definitions are often simpler to state ('multiply by 3') but harder to evaluate for large n (must calculate all terms before it). Explicit formulas are harder to find ('g_n = 3n−13^{n-1}') but let you jump to any term: g_100 directly without finding g_1 through g_99. Each form has trade-offs.

Question 17

The sequence ana_n is defined recursively by a1=1a_1=1 and an+1=an+na_{n+1}=a_n+n for n≥1n\ge 1 (so the input nn is an integer term number). What is a5a_5?

  1. 1010
  2. 1111 (correct answer)
  3. 1515
  4. 1616
Explanation: This question tests your understanding that sequences are special functions where the domain is a subset of the integers—meaning they're only defined for whole number inputs like 1, 2, 3, not for values in between. Recursive definitions specify how to get each term from previous term(s): here a₁=1, a_{n+1}=a_n + n for n≥1, adding the current n each time. To find a₅, compute step by step: a₂=1+1=2, a₃=2+2=4, a₄=4+3=7, a₅=7+4=11. This sums the first (k-1) positives plus 1, like triangular numbers shifted. Choice B correctly evaluates recursively as 11. Option C (15) might come from adding up to 5 instead of 4, but note the recursion adds n at step to n+1. Evaluating recursive sequences systematically: (1) Write initial a₁=1, (2) Table n=1 to 5, (3) Add the appropriate n each time, (4) Build to the term. The explicit a_n = n(n-1)/2 +1 = 11 for n=5 confirms it—compare forms!

Question 18

A sequence cc is a function with domain n∈{1,2,3,… }n \in \{1,2,3,\dots\}. It is defined recursively by c1=1c_1=1, c2=2c_2=2, and cn=cn−1+cn−2c_n=c_{n-1}+c_{n-2} for n≥3n \geq 3. What is c6c_6?

  1. c6=8c_6=8
  2. c6=13c_6=13 (correct answer)
  3. c6=11c_6=11
  4. c6=7c_6=7
Explanation: This question tests your understanding that sequences are special functions where the domain is a subset of the integers—meaning they're only defined for whole number inputs like 1, 2, 3, not for values in between. Recursive definitions specify how to get each term from previous term(s): the Fibonacci sequence is the classic example with c(1)=1c(1)=1, c(2)=2c(2)=2 and c(n)=c(n−1)+c(n−2)c(n)=c(n-1)+c(n-2) for n≥3n \geq 3, meaning each term equals the sum of the two before it. To find c(6)c(6), you can't jump there directly—you must calculate c(3)=2+1=3c(3)=2+1=3, then c(4)=3+2=5c(4)=3+2=5, c(5)=5+3=8c(5)=5+3=8, c(6)=8+5=13c(6)=8+5=13. Recursion requires building up step by step from the initial values! Let's evaluate step by step: c(1)=1c(1)=1, c(2)=2c(2)=2, c(3)=2+1=3c(3)=2+1=3, c(4)=3+2=5c(4)=3+2=5, c(5)=5+3=8c(5)=5+3=8, c(6)=8+5=13c(6)=8+5=13. Choice B correctly evaluates recursively as c(6)=13c(6)=13. A mistake like choice A=8 could be from using different initials like 1,1 and stopping at c(5)c(5), but adjust for the given starts—keep up the excellent work! Evaluating recursive sequences systematically: (1) Write down the initial value(s) clearly: c(1)=1c(1)=1, c(2)=2c(2)=2, (2) Set up a table or list: n=1 to 6, (3) Apply the recursive rule one step at a time, (4) Continue until you reach the desired term. Don't skip steps—recursion is sequential by nature!

Question 19

A sequence satisfies a1=2a_1 = 2 and an+1=an2−3an+4a_{n+1} = a_n^2 - 3a_n + 4 for n≥1n \geq 1. When viewed as a function, what is f(3)f(3) where f(n)=anf(n) = a_n?

  1. 1010
  2. 44
  3. 66
  4. 22 (correct answer)
Explanation: When you encounter a recursive sequence like this, you're working with a pattern where each term depends on the previous term according to a given formula. The key is to calculate terms step by step until you reach the desired position. Starting with a1=2a_1 = 2, you can find a2a_2 using the recursive formula an+1=an2−3an+4a_{n+1} = a_n^2 - 3a_n + 4: a2=a12−3a1+4=22−3(2)+4=4−6+4=2a_2 = a_1^2 - 3a_1 + 4 = 2^2 - 3(2) + 4 = 4 - 6 + 4 = 2 Now find a3a_3: a3=a22−3a2+4=22−3(2)+4=4−6+4=2a_3 = a_2^2 - 3a_2 + 4 = 2^2 - 3(2) + 4 = 4 - 6 + 4 = 2 Since f(n)=anf(n) = a_n, we have f(3)=a3=2f(3) = a_3 = 2. Looking at the wrong answers: Choice A (1010) might result from incorrectly applying the formula or making arithmetic errors in the calculations. Choice B (44) could come from confusing this with a12=4a_1^2 = 4 or misapplying the recursive relationship. Choice C (66) might arise from calculation mistakes, such as getting 22+3(2)=102^2 + 3(2) = 10 and then making another error. Notice something interesting here: once a2=2a_2 = 2, the sequence becomes constant because substituting 22 into the recursive formula always yields 22. This means 22 is a fixed point of the recurrence relation. Strategy tip: With recursive sequences, always work step-by-step and double-check your arithmetic. Also watch for patterns—sometimes sequences stabilize, repeat, or follow predictable behaviors that can save you calculation time.

Question 20

A sequence is defined recursively by a1=3a_1 = 3 and an+1=2an−1a_{n+1} = 2a_n - 1 for n≥1n \geq 1. If this sequence represents a function ff where f(n)=anf(n) = a_n, what is the domain of this function?

  1. All real numbers
  2. All positive integers starting from 1 (correct answer)
  3. All integers greater than or equal to 0
  4. All real numbers greater than or equal to 1
Explanation: Since sequences are functions whose domain is a subset of the integers, and this sequence is defined starting at n=1 with recursive terms for n≥1, the domain consists of all positive integers {1, 2, 3, 4, ...}. Choice A is wrong because sequences don't have real number domains. Choice C is wrong because the sequence starts at n=1, not n=0. Choice D is wrong because the domain must be integers, not all real numbers ≥1.