Sequences as Functions and Recursion - Algebra 1

A sequence is defined recursively by $a_1=6$ and $a_{n+1}=\frac{1}{2}a_n$ for $n=1,2,3,\dots$ . What is $a_4$ ?
Answer: $\frac{3}{4}$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A recursive definition tells you how to get each term from the previous term(s): it gives you a starting value (like a₁ = 6) and a rule for finding the next one (like aₙ₊₁ = ½aₙ, which means 'multiply the previous term by ½'). Let's find a₄ step by step using the recursive definition. We start with a₁ = 6. Then we apply the rule aₙ₊₁ = ½aₙ: a₂ = ½·a₁ = ½·6 = 3, then a₃ = ½·a₂ = ½·3 = 3/2, and finally a₄ = ½·a₃ = ½·(3/2) = 3/4. So a₄ = 3/4. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice B is correct because it correctly evaluates each term following the recursive rule to get 3/4. Great work following the pattern! Choice C (3/2) is actually a₃, not a₄—it stops one step too early. When following recursive rules, each step has to follow the exact rule—one small change throws off all the later terms! When working with recursively defined sequences, always start by writing down the initial term(s) clearly, then apply the rule one step at a time, calculating each new term before moving to the next. It's like following a recipe—don't skip steps! If you need the 4th term, you'll typically need to find the 2nd and 3rd terms along the way.
The population follows the rule $P(0) = 100$ and $P(t) = 1.5 \cdot P(t-1) + 50$ for $t \geq 1$ . After how many complete hours will the population first exceed 500 bacteria?
Answer: After 3 complete hours the population exceeds 500 We calculate the population hour by hour: P(0) = 100, P(1) = 1.5(100) + 50 = 200, P(2) = 1.5(200) + 50 = 350, P(3) = 1.5(350) + 50 = 575. Since P(3) = 575 > 500, the population first exceeds 500 after 3 complete hours. Choice B (4 hours) would be chosen by students who miscalculate P(3) or think we need P(4). Choice C (5 hours) might result from errors in the recursive calculation. Choice D (6 hours) would result from systematic calculation errors or misunderstanding the growth rate.
A sequence $a(n)$ satisfies $a(1) = 2$ and $a(n) = a(n-1) + 2n - 1$ for $n \geq 2$ . Which expression represents $a(4)$ in terms of the initial value?
Answer: $a(4) = 2 + (2 \cdot 2 - 1) + (2 \cdot 3 - 1) + (2 \cdot 4 - 1)$ When you encounter a recursive sequence problem, you need to carefully track how each term builds from the previous one using the given rule. This question tests your ability to expand a recursive formula step by step. Starting with $a(1) = 2$ and the rule $a(n) = a(n-1) + 2n - 1$ , let's build each term: - $a(2) = a(1) + (2 \cdot 2 - 1) = 2 + 3$ - $a(3) = a(2) + (2 \cdot 3 - 1) = a(1) + (2 \cdot 2 - 1) + (2 \cdot 3 - 1)$ - $a(4) = a(3) + (2 \cdot 4 - 1) = a(1) + (2 \cdot 2 - 1) + (2 \cdot 3 - 1) + (2 \cdot 4 - 1)$ This gives us $a(4) = 2 + (2 \cdot 2 - 1) + (2 \cdot 3 - 1) + (2 \cdot 4 - 1)$ , which is answer choice B. Choice A incorrectly includes an extra term $(2 \cdot 5 - 1)$ , which would be needed for $a(5)$ , not $a(4)$ . Choice C starts the added terms at $n = 1$ instead of $n = 2$ —remember that the recursive rule only applies for $n \geq 2$ , so the first term you add to $a(1)$ corresponds to $n = 2$ . Choice D incorrectly suggests multiplying one term by 3 instead of adding three separate terms. Study tip: In recursive sequence problems, write out the first few terms explicitly and look for the pattern. Count carefully—to get from $a(1)$ to $a(4)$ , you need exactly 3 applications of the recursive rule.
The sequence $2, 6, 18, 54, \dots$ can be viewed as a function with input $n = 1,2,3,\dots$ and output $a_n$ . Which recursive rule generates this sequence?
Answer: $a_1=2,\ a_{n+1}=3a_n$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A sequence is just a function where the inputs are whole numbers (like 1, 2, 3, ...) and each output is called a term: the first term a₁, the second term a₂, and so on. We can think of a sequence as a list of numbers where each number's position (1st, 2nd, 3rd) is its input! Looking at the sequence 2, 6, 18, 54, ..., let's find the pattern between consecutive terms: from 2 to 6, we multiply by 3. From 6 to 18, we multiply by 3. This pattern holds for all consecutive terms, so the recursive rule is aₙ₊₁ = 3aₙ, with starting value a₁ = 2. Choice D is correct because it properly identifies the pattern as multiplying by 3 each time with the correct starting value. Choice C applies the wrong operation in the recursive rule, multiplying by 2 when the pattern shows we should multiply by 3. Look carefully at how consecutive terms relate: are we adding the same amount? Multiplying? Doing something else? To write a recursive rule from a sequence, compare consecutive terms: What's happening from term to term? Are we adding the same number (arithmetic)? Multiplying by the same number (geometric)? Adding the previous two (Fibonacci-style)? Once you spot the pattern, write it as aₙ₊₁ = [rule using aₙ], and don't forget to state your starting value! Remember: sequences ARE functions, just special ones! The input is always a positive integer (the term number), and the output is the term value.
A sequence $(a_n)$ has domain $n=1,2,3,\dots$ and begins $9, 6, 3, 0, \dots$ . Which recursive definition generates this sequence?
Answer: $a_1=9,\ a_{n+1}=a_n-3$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A recursive definition tells you how to get each term from the previous term(s): it gives you a starting value (like a₁ = 9) and a rule for finding the next one (like aₙ₊₁ = aₙ - 3, which means 'subtract 3 from the previous term'). Looking at the sequence 9, 6, 3, 0, ..., let's find the pattern between consecutive terms: from 9 to 6, we subtract 3. From 6 to 3, we subtract 3. From 3 to 0, we subtract 3. This pattern holds for all consecutive terms, so the recursive rule is aₙ₊₁ = aₙ - 3, with starting value a₁ = 9. Choice A is correct because it properly identifies the pattern as subtracting 3 each time and starts with the correct initial value of 9. Great work following the pattern! Choice B has the right starting value but uses the wrong operation, adding 3 when the pattern shows we should subtract 3. Look carefully at how consecutive terms relate: are we adding the same amount? Multiplying? Doing something else? To write a recursive rule from a sequence, compare consecutive terms: What's happening from term to term? Are we adding the same number (arithmetic)? Multiplying by the same number (geometric)? Adding the previous two (Fibonacci-style)? Once you spot the pattern, write it as aₙ₊₁ = [rule using aₙ], and don't forget to state your starting value!
A sequence is defined by the explicit formula $a_n=12-2n$ for $n=1,2,3,\dots$ . What is $a_4$ ?
Answer: $4$ This question tests your understanding that sequences are special functions with integer domains, and how to work with explicitly defined sequences. A sequence is just a function where the inputs are whole numbers (like 1, 2, 3, ...) and each output is called a term: the first term a₁, the second term a₂, and so on. We can think of a sequence as a list of numbers where each number's position (1st, 2nd, 3rd) is its input! Unlike recursive definitions, an explicit formula like aₙ = 12 - 2n lets us find any term directly by plugging in the value of n. To find a₄, we simply substitute n = 4 into the formula: a₄ = 12 - 2(4) = 12 - 8 = 4. So the 4th term is 4. Choice C is correct because it properly evaluates the explicit formula with n = 4 to get 4. Great work following the pattern! Choice D gives 8, which might come from calculating 12 - 4 = 8 instead of 12 - 2(4) = 4. Remember to multiply 2 times n before subtracting from 12! The beauty of explicit formulas is that you can jump directly to any term—no need to find all the terms before it! Notice that this sequence decreases by 2 each time: a₁ = 10, a₂ = 8, a₃ = 6, a₄ = 4, and so on.
A sequence is defined as a function on $n = 1, 2, 3, \dots$ by $a_1 = 6$ and $a_{n+1} = a_n - 1$ . What is $a_6$ ?
Answer: $1$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A recursive definition tells you how to get each term from the previous term(s): it gives you a starting value (like a₁ = 6) and a rule for finding the next one (like aₙ₊₁ = aₙ - 1, which means 'subtract 1 from the previous term'). Let's find a₆ step by step using the recursive definition. We start with a₁ = 6. Then we apply the rule aₙ₊₁ = aₙ - 1: a₂ = a₁ - 1 = 6 - 1 = 5, then a₃ = a₂ - 1 = 5 - 1 = 4, then a₄ = a₃ - 1 = 4 - 1 = 3, then a₅ = a₄ - 1 = 3 - 1 = 2, and finally a₆ = a₅ - 1 = 2 - 1 = 1. So a₆ = 1. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice B is correct because it correctly evaluates each term following the recursive rule to get 1. Great work following the pattern! Choice C gives 2, which is actually a₅, not a₆. When finding the nth term recursively, count carefully—you need to apply the rule (n-1) times starting from a₁! When working with recursively defined sequences, always start by writing down the initial term(s) clearly, then apply the rule one step at a time, calculating each new term before moving to the next. This sequence decreases by 1 each time, creating a simple countdown pattern. If we continued, we'd eventually reach 0, then negative numbers!
A function $f$ is defined on $n=0,1,2,\dots$ by $f(0)=1$ , $f(1)=1$ , and $f(n)=f(n-1)+f(n-2)$ for $n\ge 2$ . What is $f(6)$ ?
Answer: $13$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. The Fibonacci sequence is a famous recursive sequence where each term is the sum of the two previous terms: f(1) = 1, f(2) = 1, and f(n) = f(n-1) + f(n-2) for n ≥ 3. So f(3) = f(2) + f(1) = 1 + 1 = 2, then f(4) = f(3) + f(2) = 2 + 1 = 3, and so on—each new term builds on the ones before it! For Fibonacci-style sequences where each term is based on the previous two, we need two starting values: f(0) = 1 and f(1) = 1. Then for each new term, we add the previous two: f(2) = f(1) + f(0) = 1 + 1 = 2, f(3) = f(2) + f(1) = 2 + 1 = 3, f(4) = f(3) + f(2) = 3 + 2 = 5, f(5) = f(4) + f(3) = 5 + 3 = 8, and f(6) = f(5) + f(4) = 8 + 5 = 13. It's like each term 'inherits' from its two parents! Choice B is correct because it properly follows the Fibonacci rule, adding the previous two terms at each step to get 13. Great work following the pattern! Choice A gives 8, which is actually f(5), not f(6). With recursive sequences, one arithmetic slip affects all the terms after it! For Fibonacci-type sequences where each term combines previous terms, make a little table as you go: write n = 0, 1, 2, 3, ... down one side and the term values across the top. This helps you keep track of what you've calculated and makes it easy to reference previous terms.
The sequence is defined recursively by $a_1=10$ and $a_{n+1}=a_n-3$ for $n=1,2,3,\dots$ . What is the next term after $10, 7, 4$ ?
Answer: $1$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A recursive definition tells you how to get each term from the previous term(s): it gives you a starting value (like a₁ = 10) and a rule for finding the next one (like aₙ₊₁ = aₙ - 3, which means 'subtract 3 from the previous term'). To find any term, you start at the beginning and apply the rule step by step. The sequence starts with 10, 7, 4, so the next term after 4 is found by applying the rule: 4 - 3 = 1. So the next term is 1. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice A is correct because it correctly evaluates the next term following the recursive rule. Choice D applies the wrong operation in the recursive rule, subtracting 5 or something to get -2 when the pattern shows we should subtract 3. Look carefully at how consecutive terms relate: are we adding the same amount? Multiplying? Doing something else? When working with recursively defined sequences, always start by writing down the initial term(s) clearly, then apply the rule one step at a time, calculating each new term before moving to the next. It's like following a recipe—don't skip steps!
A sequence is defined recursively by $a_1=4$ and $a_{n+1}=a_n+3$ for $n=1,2,3,\dots$ . What is $a_5$ ?
Answer: $16$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A recursive definition tells you how to get each term from the previous term(s): it gives you a starting value (like a₁ = 4) and a rule for finding the next one (like aₙ₊₁ = aₙ + 3, which means 'add 3 to the previous term'). Let's find a₅ step by step using the recursive definition. We start with a₁ = 4. Then we apply the rule aₙ₊₁ = aₙ + 3: a₂ = a₁ + 3 = 4 + 3 = 7, then a₃ = a₂ + 3 = 7 + 3 = 10, then a₄ = a₃ + 3 = 10 + 3 = 13, and finally a₅ = a₄ + 3 = 13 + 3 = 16. So a₅ = 16. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice B is correct because it correctly evaluates each term following the recursive rule to get 16. Great work following the pattern! Choice A (13) is actually a₄, not a₅—it stops one step too early. When finding a specific term, make sure to count carefully and continue all the way to the term you need! When working with recursively defined sequences, always start by writing down the initial term(s) clearly, then apply the rule one step at a time, calculating each new term before moving to the next. It's like following a recipe—don't skip steps! If you need the 5th term, you'll typically need to find the 2nd, 3rd, and 4th terms along the way.
A sequence is defined as a function $f$ on $n=1,2,3,\dots$ by $f(1)=2$ and $f(n+1)=f(n)+f(n)$ (double the previous value). What are the first four terms $f(1), f(2), f(3), f(4)$ ?
Answer: $2,\ 4,\ 8,\ 16$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A recursive definition tells you how to get each term from the previous term(s): it gives you a starting value (like f(1) = 2) and a rule for finding the next one (like f(n+1) = f(n) + f(n), which means 'double the previous term'). Let's find the first four terms step by step using the recursive definition. We start with f(1) = 2. Then we apply the rule f(n+1) = f(n) + f(n) = 2f(n): f(2) = f(1) + f(1) = 2 + 2 = 4, then f(3) = f(2) + f(2) = 4 + 4 = 8, and f(4) = f(3) + f(3) = 8 + 8 = 16. So the first four terms are 2, 4, 8, 16. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice B is correct because it correctly evaluates each term following the recursive rule to get 2, 4, 8, 16. Great work following the pattern! Choice A makes an error by adding 2 instead of doubling, getting 2, 4, 6, 8. When following recursive rules, each step has to follow the exact rule—one small change throws off all the later terms! When working with recursively defined sequences, always start by writing down the initial term(s) clearly, then apply the rule one step at a time, calculating each new term before moving to the next. It's like following a recipe—don't skip steps! The beauty of recursive definitions is that they describe the pattern in the simplest way: 'start at 2, double each time' is clear and natural.
A sequence is defined recursively by $a_1=1$ , $a_2=1$ , and $a_n=a_{n-1}+a_{n-2}$ for $n \geq 3$ (domain $n=1,2,3,\dots$ ). What is $a_6$ ?
Answer: $8$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. The Fibonacci sequence is a famous recursive sequence where each term is the sum of the two previous terms: $a_1 = 1$ , $a_2 = 1$ , and $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$ . So $a_3 = a_2 + a_1 = 1 + 1 = 2$ , then $a_4 = a_3 + a_2 = 2 + 1 = 3$ , and so on—each new term builds on the ones before it! For Fibonacci-style sequences where each term is based on the previous two, we need two starting values: $a_1 = 1$ and $a_2 = 1$ . Then for each new term, we add the previous two: $a_3 = 1 + 1 = 2$ , $a_4 = 2 + 1 = 3$ , $a_5 = 3 + 2 = 5$ , $a_6 = 5 + 3 = 8$ . It's like each term 'inherits' from its two parents! Choice B is correct because it accurately calculates to get 8. Choice C gets the first few terms right but makes an arithmetic error when calculating $a_6$ , computing $5 + 8 = 13$ instead of $5 + 3 = 8$ . With recursive sequences, one arithmetic slip affects all the terms after it! For Fibonacci-type sequences where each term combines previous terms, make a little table as you go: write $n = 1, 2, 3, \dots$ down one side and the term values across the top. This helps you keep track of what you've calculated and makes it easy to reference previous terms.
The sequence $1, 1, 2, 3, 5, \dots$ is defined by $f(1)=1$ , $f(2)=1$ , and $f(n)=f(n-1)+f(n-2)$ for $n\ge 3$ (with domain $n=1,2,3,\dots$ ). What is $f(6)$ ?
Answer: $8$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. The Fibonacci sequence is a famous recursive sequence where each term is the sum of the two previous terms: f(1) = 1, f(2) = 1, and f(n) = f(n-1) + f(n-2) for n ≥ 3. So f(3) = f(2) + f(1) = 1 + 1 = 2, then f(4) = f(3) + f(2) = 2 + 1 = 3, and so on—each new term builds on the ones before it! For Fibonacci-style sequences where each term is based on the previous two, we need two starting values: f(1) = 1 and f(2) = 1. Then for each new term, we add the previous two: f(3) = f(2) + f(1) = 1 + 1 = 2, f(4) = f(3) + f(2) = 2 + 1 = 3, f(5) = f(4) + f(3) = 3 + 2 = 5, and f(6) = f(5) + f(4) = 5 + 3 = 8. It's like each term 'inherits' from its two parents! Choice B is correct because it correctly calculates f(6) = 8 by following the Fibonacci rule. Great work following the pattern! Choice A (5) is actually f(5), not f(6)—it stops one step too early. When finding a specific term, make sure to count carefully and continue all the way to the term you need! For Fibonacci-type sequences where each term combines previous terms, make a little table as you go: write n = 1, 2, 3, ... down one side and the term values across the top. This helps you keep track of what you've calculated and makes it easy to reference previous terms.
A sequence is defined recursively for $n = 1, 2, 3, \dots$ by $a_1 = 10$ and $a_{n+1} = a_n - 2$ . What are the first four terms $a_1, a_2, a_3, a_4$ ?
Answer: $10, 8, 6, 4$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A recursive definition tells you how to get each term from the previous term(s): it gives you a starting value (like a₁ = 10) and a rule for finding the next one (like aₙ₊₁ = aₙ - 2, which means 'subtract 2 from the previous term'). Let's find the first four terms step by step using the recursive definition. We start with a₁ = 10. Then we apply the rule aₙ₊₁ = aₙ - 2: a₂ = a₁ - 2 = 10 - 2 = 8, then a₃ = a₂ - 2 = 8 - 2 = 6, and finally a₄ = a₃ - 2 = 6 - 2 = 4. So the first four terms are 10, 8, 6, 4. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice A is correct because it correctly evaluates each term following the recursive rule to get 10, 8, 6, 4. Great work following the pattern! Choice C makes an error in the third step: it shows 10, 8, 7, 6, which suggests subtracting 2, then 1, then 1—but the rule says to subtract 2 every time. When following recursive rules, each step has to follow the exact rule—one small change throws off all the later terms! When working with recursively defined sequences, always start by writing down the initial term(s) clearly, then apply the rule one step at a time, calculating each new term before moving to the next. It's like following a recipe—don't skip steps! The beauty of recursive definitions is that they describe the pattern in the simplest way: 'start at 10, subtract 2 each time' is clear and natural.
A function $f$ is defined on the integers $n=1,2,3,\dots$ by $f(1)=8$ and $f(n+1)=f(n)-1$ . What are $f(1), f(2), f(3), f(4)$ ?
Answer: $8,\ 7,\ 6,\ 5$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A recursive definition tells you how to get each term from the previous term(s): it gives you a starting value (like f(1) = 8) and a rule for finding the next one (like f(n+1) = f(n) - 1, which means 'subtract 1 from the previous term'). Let's find f(1), f(2), f(3), f(4) step by step using the recursive definition. We start with f(1) = 8. Then we apply the rule f(n+1) = f(n) - 1: f(2) = f(1) - 1 = 8 - 1 = 7, then f(3) = f(2) - 1 = 7 - 1 = 6, and finally f(4) = f(3) - 1 = 6 - 1 = 5. So the first four terms are 8, 7, 6, 5. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice A is correct because it correctly evaluates each term following the recursive rule to get 8, 7, 6, 5. Great work following the pattern! Choice B adds 1 instead of subtracting 1. When following recursive rules, each step has to follow the exact rule—one small change throws off all the later terms! When working with recursively defined sequences, always start by writing down the initial term(s) clearly, then apply the rule one step at a time, calculating each new term before moving to the next. Remember: sequences ARE functions, just special ones! The input is always a positive integer (the term number), and the output is the term value.
The sequence $2, 6, 18, 54, \dots$ is a function with domain $n=1,2,3,\dots$ where $a_n$ is the $n$ th term. Which recursive definition matches the sequence?
Answer: $a_1=2,\ a_{n+1}=3a_n$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A sequence is just a function where the inputs are whole numbers (like 1, 2, 3, ...) and each output is called a term: the first term a₁, the second term a₂, and so on. We can think of a sequence as a list of numbers where each number's position (1st, 2nd, 3rd) is its input! Looking at the sequence 2, 6, 18, 54, ..., let's find the pattern between consecutive terms: from 2 to 6, we multiply by 3 (2 × 3 = 6). From 6 to 18, we multiply by 3 (6 × 3 = 18). From 18 to 54, we multiply by 3 (18 × 3 = 54). This pattern holds for all consecutive terms, so the recursive rule is aₙ₊₁ = 3aₙ, with starting value a₁ = 2. Choice C is correct because it properly identifies the pattern as multiplying by 3 each time and starts with the correct initial value of 2. Great work following the pattern! Choice B multiplies by 2 instead of 3. Look carefully at how consecutive terms relate: 2 × 2 = 4, not 6, so this can't be the right rule! To write a recursive rule from a sequence, compare consecutive terms: What's happening from term to term? Are we adding the same number (arithmetic)? Multiplying by the same number (geometric)? Once you spot the pattern, write it as aₙ₊₁ = [rule using aₙ], and don't forget to state your starting value!
A function $f$ is defined on the integers $n=1,2,3,\dots$ by $f(1)=2$ and $f(n+1)=3\,f(n)$ for $n\ge 1$ . What is $f(4)$ ?
Answer: $54$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A recursive definition tells you how to get each term from the previous term(s): it gives you a starting value (like f(1) = 2) and a rule for finding the next one (like f(n+1) = 3f(n), which means 'multiply the previous term by 3'). Let's find f(4) step by step using the recursive definition. We start with f(1) = 2. Then we apply the rule f(n+1) = 3f(n): f(2) = 3·f(1) = 3·2 = 6, then f(3) = 3·f(2) = 3·6 = 18, and finally f(4) = 3·f(3) = 3·18 = 54. So f(4) = 54. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice B is correct because it properly follows the recursive rule, multiplying by 3 at each step to get 54. Great work following the pattern! Choice A makes an error by getting 18, which is actually f(3), not f(4). When following recursive rules, each step has to follow the exact rule—one small change throws off all the later terms! When working with recursively defined sequences, always start by writing down the initial term(s) clearly, then apply the rule one step at a time, calculating each new term before moving to the next. For geometric sequences like this one where we multiply by the same number each time, the values grow very quickly!
A sequence is defined recursively by $a_1=3$ and $a_{n+1}=a_n+4$ for $n=1,2,3,\dots$ . What are the first four terms $a_1, a_2, a_3, a_4$ ?
Answer: $3,\ 7,\ 11,\ 15$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A recursive definition tells you how to get each term from the previous term(s): it gives you a starting value (like a₁ = 3) and a rule for finding the next one (like aₙ₊₁ = aₙ + 4, which means 'add 4 to the previous term'). To find any term, you start at the beginning and apply the rule step by step. Let's find the first four terms step by step using the recursive definition. We start with a₁ = 3. Then we apply the rule aₙ₊₁ = aₙ + 4: a₂ = 3 + 4 = 7, then a₃ = 7 + 4 = 11, and a₄ = 11 + 4 = 15. So the first four terms are 3, 7, 11, 15. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice A is correct because it correctly evaluates each term following the recursive rule. Choice C has the right pattern but starts with the wrong initial value, using 7 instead of 3. The starting value is crucial in recursive sequences—like the first domino that sets everything else in motion! When working with recursively defined sequences, always start by writing down the initial term(s) clearly, then apply the rule one step at a time, calculating each new term before moving to the next. It's like following a recipe—don't skip steps!
A function $f$ is defined on the integers $n = 0, 1, 2, 3, \dots$ by $f(0) = 5$ and $f(n+1) = f(n) + 4$ . What is $f(3)$ ?
Answer: $17$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A recursive definition tells you how to get each term from the previous term(s): it gives you a starting value (like f(0) = 5) and a rule for finding the next one (like f(n+1) = f(n) + 4, which means 'add 4 to the previous term'). Let's find f(3) step by step using the recursive definition. We start with f(0) = 5. Then we apply the rule f(n+1) = f(n) + 4: f(1) = f(0) + 4 = 5 + 4 = 9, then f(2) = f(1) + 4 = 9 + 4 = 13, and finally f(3) = f(2) + 4 = 13 + 4 = 17. So f(3) = 17. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice C is correct because it correctly evaluates each term following the recursive rule to get 17. Great work following the pattern! Choice B gives 13, which is actually f(2), not f(3). This is a common error when the domain starts at n = 0 instead of n = 1—make sure to count your steps carefully from the starting point! When working with recursively defined sequences, always start by writing down the initial term(s) clearly, then apply the rule one step at a time, calculating each new term before moving to the next. Notice that this sequence starts at n = 0, not n = 1—always check where your domain begins!
A sequence is defined recursively by $a_1=8$ and $a_{n+1}=\dfrac{1}{2}a_n$ for $n=1,2,3,\dots$ . What is $a_4$ ?
Answer: $1$ This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A recursive definition tells you how to get each term from the previous term(s): it gives you a starting value (like a₁ = 8) and a rule for finding the next one (like aₙ₊₁ = ½aₙ, which means 'multiply the previous term by ½' or 'divide by 2'). Let's find a₄ step by step using the recursive definition. We start with a₁ = 8. Then we apply the rule aₙ₊₁ = ½aₙ: a₂ = ½·a₁ = ½·8 = 4, then a₃ = ½·a₂ = ½·4 = 2, and finally a₄ = ½·a₃ = ½·2 = 1. So a₄ = 1. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice A is correct because it properly follows the recursive rule, dividing by 2 at each step to get 1. Great work following the pattern! Choice B gives 2, which is actually a₃, not a₄. When following recursive rules, make sure you count your steps carefully—it's easy to stop one term too early! When working with recursively defined sequences, always start by writing down the initial term(s) clearly, then apply the rule one step at a time. This is a geometric sequence where each term is half the previous one—the terms get smaller and smaller, approaching zero.

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