Sequences as Functions and Recursion - Algebra 2

Let $a$ be a sequence (discrete function) with domain $n\in\{1,2,3,\dots\}$ defined by $a_1=5$ and $a_{n+1}=a_n+7$ for $n\ge 1$ . What is $a_5$ ?
Answer: $a_5=33$ 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): this is an arithmetic sequence with a(1)=5 and a(n+1)=a(n)+7, adding a constant each time. To find a(5), build step by step or recognize the pattern. Let's evaluate step by step: a(1)=5, a(2)=5+7=12, a(3)=12+7=19, a(4)=19+7=26, a(5)=26+7=33. Choice B correctly evaluates recursively as a(5)=33. A mistake like choice A=26 could be stopping at a(4), but keep going to the fifth term—nice effort! Evaluating recursive sequences systematically: (1) Write down the initial value: a(1)=5, (2) Set up a list, (3) Add 7 each time, (4) Continue to n=5. Recursive vs explicit: here explicit is $a_n$ =5 + 7(n-1), letting you jump to $a_5$ =33 directly—each has advantages!
Two sequences are defined as follows: Sequence A: $u_1 = 4$ , $u_n = u_{n-1} + 3$ for $n \geq 2$ . Sequence B: $v_1 = 4$ , $v_n = 3v_{n-1}$ for $n \geq 2$ . When comparing these sequences as functions, which statement is true?
Answer: Sequence A can be written as $u(n) = 3n + 1$ and Sequence B grows exponentially 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.
A population model is defined by $P_0 = 100$ and $P_n = 1.2P_{n-1} - 5$ for $n \geq 1$ . If we interpret this as a function $g(n) = P_n$ , what is $g(3) - g(1)$ ?
Answer: 31.6 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.
A sequence is defined recursively by $a_1 = 3$ and $a_{n+1} = 2a_n - 1$ for $n \geq 1$ . If this sequence represents a function $f$ where $f(n) = a_n$ , what is the domain of this function?
Answer: All positive integers starting from 1 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.
A sequence $c$ is a function with domain $n \in \{1,2,3,\dots\}$ . It is defined recursively by $c_1=1$ , $c_2=2$ , and $c_n=c_{n-1}+c_{n-2}$ for $n \geq 3$ . What is $c_6$ ?
Answer: $c_6=13$ 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 $c(1)=1$ , $c(2)=2$ and $c(n)=c(n-1)+c(n-2)$ for $n \geq 3$ , meaning each term equals the sum of the two before it. To find $c(6)$ , you can't jump there directly—you must calculate $c(3)=2+1=3$ , then $c(4)=3+2=5$ , $c(5)=5+3=8$ , $c(6)=8+5=13$ . Recursion requires building up step by step from the initial values! Let's evaluate step by step: $c(1)=1$ , $c(2)=2$ , $c(3)=2+1=3$ , $c(4)=3+2=5$ , $c(5)=5+3=8$ , $c(6)=8+5=13$ . Choice B correctly evaluates recursively as $c(6)=13$ . A mistake like choice A=8 could be from using different initials like 1,1 and stopping at $c(5)$ , but adjust for the given starts—keep up the excellent work! Evaluating recursive sequences systematically: (1) Write down the initial value(s) clearly: $c(1)=1$ , $c(2)=2$ , (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!
The sequence $1,3,9,27,81,\dots$ can be treated as a function $g(n)$ with domain $n\in\{1,2,3,\dots\}$ . Which recursive definition correctly generates this sequence?
Answer: $g_1=1,\; g_{n+1}=3g_n$ 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)=3*1=3, g(3)=3*3=9, g(4)=3*9=27, g(5)=3*27=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 (' $g_n$ = $3^{n-1}$ ') but let you jump to any term: $g_{100}$ directly without finding $g_1$ through $g_{99}$ . Each form has trade-offs.
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 $a_1, a_2, a_3, \dots$ ?
Answer: All positive integers $\{1,2,3,\dots\}$ 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!
A sequence can be viewed as a function whose domain is a subset of the integers. For the Fibonacci sequence defined on $n \in \{1,2,3,\dots\}$ by $f(1)=1$ , $f(2)=1$ , and $f(n)=f(n-1)+f(n-2)$ for $n\ge 3$ , find $f(8)$ .
Answer: $21$ 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 $f(1) = 1$ , $f(2) = 1$ and $f(n) = f(n-1) + f(n-2)$ , meaning each term equals the sum of the two before it. To find $f(8)$ , start with $f(1)=1$ , $f(2)=1$ , then $f(3)=2$ , $f(4)=3$ , $f(5)=5$ , $f(6)=8$ , $f(7)=13$ , and $f(8)=21$ —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 $f(7)$ instead of $f(8)$ , so always verify the term number. 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, 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!
Consider the recursively defined sequence $b_0 = 5$ , $b_1 = 2$ , and $b_n = b_{n-1} + b_{n-2}$ for $n \geq 2$ . Which statement about this sequence as a function is most accurate?
Answer: The function requires exactly two previous terms to determine each subsequent value 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.
A sequence is a function defined only at integer inputs. Which set could be the domain of a sequence $a(n)$ (viewed as a discrete function)?
Answer: $\{1,2,3,4,\dots\}$ 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.
A sequence can be viewed as a discrete function $f$ whose domain is the positive integers $n \in \{1,2,3,\dots\}$ . For the Fibonacci sequence defined by $f(1)=f(2)=1$ and $f(n)=f(n-1)+f(n-2)$ for $n\ge 3$ , what is $f(7)$ ?
Answer: $13$ 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!
The sequence $a_n$ is defined recursively by $a_1=1$ and $a_{n+1}=a_n+n$ for $n\ge 1$ (so the input $n$ is an integer term number). What is $a_5$ ?
Answer: $11$ 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!
A sequence satisfies $a_1 = 2$ and $a_{n+1} = a_n^2 - 3a_n + 4$ for $n \geq 1$ . When viewed as a function, what is $f(3)$ where $f(n) = a_n$ ?
Answer: 2 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 $a_1 = 2$ , you can find $a_2$ using the recursive formula $a_{n+1} = a_n^2 - 3a_n + 4$ : $a_2 = a_1^2 - 3a_1 + 4 = 2^2 - 3(2) + 4 = 4 - 6 + 4 = 2$ Now find $a_3$ : $a_3 = a_2^2 - 3a_2 + 4 = 2^2 - 3(2) + 4 = 4 - 6 + 4 = 2$ Since $f(n) = a_n$ , we have $f(3) = a_3 = 2$ . Looking at the wrong answers: Choice A (10) might result from incorrectly applying the formula or making arithmetic errors in the calculations. Choice B (4) could come from confusing this with $a_1^2 = 4$ or misapplying the recursive relationship. Choice C (6) might arise from calculation mistakes, such as getting $2^2 + 3(2) = 10$ and then making another error. Notice something interesting here: once $a_2 = 2$ , the sequence becomes constant because substituting 2 into the recursive formula always yields 2. This means 2 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.
The sequence $1,3,9,27,81,\dots$ is a function with domain $n\in\{1,2,3,\dots\}$ . Which recursive definition matches this sequence?
Answer: $a_1=1,\ a_{n+1}=3a_n$ 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 = 3*1 = 3, a₃ = 3*3 = 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.
A sequence is defined recursively where $c_1 = 5$ and $c_n = (-1)^n \cdot c_{n-1} + 2$ for $n \geq 2$ . If we consider the function $f(n) = c_n$ , what is the relationship between $f(4)$ and $f(6)$ ?
Answer: $f(4) = f(6)$ 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 $n$ is even or odd. Let's calculate the terms step by step using $c_n = (-1)^n \cdot c_{n-1} + 2$ : $c_1 = 5$ (given) $c_2 = (-1)^2 \cdot c_1 + 2 = 1 \cdot 5 + 2 = 7$ $c_3 = (-1)^3 \cdot c_2 + 2 = -1 \cdot 7 + 2 = -5$ $c_4 = (-1)^4 \cdot c_3 + 2 = 1 \cdot (-5) + 2 = -3$ $c_5 = (-1)^5 \cdot c_4 + 2 = -1 \cdot (-3) + 2 = 5$ $c_6 = (-1)^6 \cdot c_5 + 2 = 1 \cdot 5 + 2 = 7$ Notice that $c_5 = c_1 = 5$ and $c_6 = c_2 = 7$ . The sequence repeats every 4 terms! Therefore $f(4) = c_4 = -3$ and $f(6) = c_6 = -3$ , making choice D correct. Choice A suggests $f(4) = f(6) - 4$ , which would mean $-3 = -3 - 4 = -7$ , which is false. Choice B claims $f(4) = -f(6)$ , meaning $-3 = -(-3) = 3$ , also incorrect. Choice C proposes $f(4) = f(6) + 4$ , giving us $-3 = -3 + 4 = 1$ , 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.
Two students define sequences differently. Student A: $x_n = 2n + 1$ for $n \geq 1$ . Student B: $y_1 = 3$ and $y_n = y_{n-1} + 2$ for $n \geq 2$ . Which statement about these sequences as functions is most accurate?
Answer: Both sequences represent the same function with domain of positive integers 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.
Consider the sequence $t_1 = 8$ , $t_2 = 12$ , and $t_n = \frac{t_{n-1} + t_{n-2}}{2}$ for $n \geq 3$ . If this represents a function $h(n) = t_n$ , which property best describes the long-term behavior?
Answer: The function values approach a finite limit as $n$ approaches infinity 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.
A sequence can be viewed as a discrete function whose domain is a subset of the integers. For the Fibonacci sequence defined by $f(1)=f(2)=1$ and $f(n)=f(n-1)+f(n-2)$ for $n\ge 3$ (with domain $n\in\{1,2,3,\dots\}$ ), find $f(7)$ .
Answer: $f(7)=13$ 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!
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 $a_1, a_2, a_3, \dots$ ?
Answer: All positive integers $\{1,2,3,\dots\}$ 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!
The sequence $2, 5, 8, 11, \dots$ is a function with domain $n \in \{1,2,3,\dots\}$ . Which recursive definition matches this sequence?
Answer: $a_1=2,\ a_{n+1}=a_n+3$ 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 $(1,2)$ , $(2,5)$ , $(3,8)$ , not a connected curve. Choice B correctly defines with initial value $a_1=2$ and rule $a_{n+1}=a_n+3$ , generating $2,5,8,11$ perfectly. Choice A fails by using +2, giving $2,4,6$ which doesn't match—remember to check the common difference of 3. Evaluating recursive sequences systematically: (1) Write down the initial value(s), (2) Set up a table, (3) Apply the rule step by step, (4) Continue to verify—don't skip steps! Recursive vs explicit: recursive is simple ('add 3') but requires steps for large n, while explicit $a_n=2+3(n-1)$ jumps directly—each has trade-offs.

Start this quiz