A function is defined on the integers by and for . What is ?
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Algebra Quiz
Practice Sequences As Functions And Recursion in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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A function f is defined on the integers n=1,2,3,… by f(1)=2 and f(n+1)=3f(n) for n≥1. What is f(4)?
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.
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.
A function f is defined on the integers n=1,2,3,… by f(1)=2 and f(n+1)=3f(n) for n≥1. What is f(4)?
Explanation: 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!
The sequence an is a function with domain n=1,2,3,… and starts 6,3,0,−3,…. Which recursive rule generates this sequence?
Explanation: 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 - 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. Looking at the sequence 6, 3, 0, -3, ..., let's find the pattern between consecutive terms: 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 = 6. Choice B is correct because it properly identifies the pattern as subtracting 3 each time with the correct starting value of 6. Great work following the pattern! Choice C has the right pattern but starts with the wrong initial value, using 3 instead of 6. The starting value is crucial in recursive sequences—like the first domino that sets everything else in motion! 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! The beauty of recursive definitions is that they describe the pattern in the simplest way: 'start at 6, subtract 3 each time' is clear and natural.
A function f is defined only for integers n=1,2,3,… by f(1)=2 and f(n+1)=2f(n). What is f(4)?
Explanation: 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) = 2f(n), which means 'multiply the previous term by 2'). To find any term, you start at the beginning and apply the rule step by step. 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) = 2f(n): f(2) = 2 * 2 = 4, then f(3) = 2 * 4 = 8, and f(4) = 2 * 8 = 16. So f(4) = 16. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice C is correct because it accurately calculates to get 16 by correctly evaluating each term following the recursive rule. Great work following the pattern! Choice D makes an error in the last step, multiplying by 2 one extra time to get 32 instead of stopping at 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. 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.
A sequence is defined recursively by a1=4 and an+1=an+3 for n=1,2,3,…. What is a5?
Explanation: 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'). To find any term, you start at the beginning and apply the rule step by step. 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=4+3=7, then a3=7+3=10, then a4=10+3=13, and a5=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 accurately calculates to get 16. Choice A makes an error in the 5th step: it stops at 13 instead of adding 3 one more 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! If you need the 5th term, you'll typically need to find the 2nd, 3rd, and 4th terms along the way.
A function f is defined on the integers n=1,2,3,… by f(1)=2 and f(n+1)=2f(n). What is f(4)?
Explanation: 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) = 2f(n), which means 'multiply the previous term by 2'). 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) = 2f(n): f(2) = 2·f(1) = 2·2 = 4, then f(3) = 2·f(2) = 2·4 = 8, and finally f(4) = 2·f(3) = 2·8 = 16. So f(4) = 16. 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 16. Great work following the pattern! Choice B makes an error in the calculation: it might have added instead of multiplied, or made an arithmetic mistake. When the recursive rule says f(n+1) = 2f(n), we multiply by 2 at each step—this creates a geometric sequence that grows quickly! 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.
A sequence is defined recursively by a1=6 and an+1=21an for n=1,2,3,…. What is a4?
Explanation: 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.
A function f is defined on the integers n=1,2,3,… by f(1)=8 and f(n+1)=f(n)−1. What are f(1),f(2),f(3),f(4)?
Explanation: 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 1,1,2,3,5,… is defined by f(1)=1, f(2)=1, and f(n)=f(n−1)+f(n−2) for n≥3 (with domain n=1,2,3,…). What is f(6)?
Explanation: 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 as a function f on the integers n=1,2,3,… by f(1)=4 and f(n+1)=f(n)+3. What is f(5)?
Explanation: 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) = 4) and a rule for finding the next one (like f(n+1) = f(n) + 3, which means 'add 3 to the previous term'). Let's find f(5) step by step using the recursive definition. We start with f(1) = 4. Then we apply the rule f(n+1) = f(n) + 3: f(2) = f(1) + 3 = 4 + 3 = 7, then f(3) = f(2) + 3 = 7 + 3 = 10, then f(4) = f(3) + 3 = 10 + 3 = 13, and finally f(5) = f(4) + 3 = 13 + 3 = 16. So f(5) = 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 C makes an error by adding 4 instead of 3 at some step, getting 19. 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 5th term, you'll typically need to find the 2nd, 3rd, and 4th terms along the way.
A function f with domain n=0,1,2,3,… is defined recursively by f(0)=3 and f(n+1)=f(n)+5. What is f(3)?
Explanation: 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) = 3) and a rule for finding the next one (like f(n+1) = f(n) + 5, which means 'add 5 to the previous term'). To find any term, you start at the beginning and apply the rule step by step. Let's find f(3) step by step using the recursive definition. We start with f(0) = 3. Then we apply the rule f(n+1) = f(n) + 5: f(1) = 3 + 5 = 8, then f(2) = 8 + 5 = 13, and f(3) = 13 + 5 = 18. So f(3) = 18. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice C is correct because it accurately calculates to get 18 by correctly evaluating each term following the recursive rule. Great work following the pattern! Choice D makes an error by stopping at f(2), giving 13 instead of continuing to add 5 for f(3). 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 3rd term, you'll typically need to find the 1st and 2nd terms along the way.
A sequence is defined as a function on n=1,2,3,… by a1=9 and an+1=31an. What is a4?
Explanation: 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 = (1/3)aₙ, which means 'multiply the previous term by 1/3'). Let's find a₄ step by step using the recursive definition. We start with a₁ = 9. Then we apply the rule aₙ₊₁ = (1/3)aₙ: a₂ = (1/3)·a₁ = (1/3)·9 = 3, then a₃ = (1/3)·a₂ = (1/3)·3 = 1, and finally a₄ = (1/3)·a₃ = (1/3)·1 = 1/3. So a₄ = 1/3. 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 1/3. Great work following the pattern! Choice A gives 1, which is actually a₃, not a₄. 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 divides by 3 at each step, creating a geometric sequence that shrinks toward zero—quite different from sequences that grow!
A function f is defined on n=0,1,2,… by f(0)=1, f(1)=1, and f(n)=f(n−1)+f(n−2) for n≥2. What is f(6)?
Explanation: 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.
A sequence is defined recursively by a1=9 and an+1=an−2 for n=1,2,3,…. What are the first four terms a1,a2,a3,a4?
Explanation: 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 - 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 = 9. Then we apply the rule aₙ₊₁ = aₙ - 2: a₂ = a₁ - 2 = 9 - 2 = 7, then a₃ = a₂ - 2 = 7 - 2 = 5, and finally a₄ = a₃ - 2 = 5 - 2 = 3. So the first four terms are 9, 7, 5, 3. 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, subtracting 2 at each step to get 9, 7, 5, 3. Great work following the pattern! Choice B shows an increasing sequence (9, 11, 13, 15) which would come from adding 2 instead of subtracting 2. 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!
A sequence is defined recursively for n=1,2,3,… by a1=10 and an+1=an−2. What are the first four terms of the sequence?
Explanation: 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 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 D has the right pattern but starts with the wrong initial value, using 8 instead of 10. 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! If you need the 4th term, you'll typically need to find the 2nd and 3rd terms along the way.
The sequence 2,6,18,54,… can be viewed as a function with input n=1,2,3,… and output an. Which recursive rule generates this sequence?
Explanation: 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 is defined recursively by a1=2 and an+1=an−5 for n=1,2,3,…. What are the first four terms a1,a2,a3,a4?
Explanation: 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 = 2) and a rule for finding the next one (like aₙ₊₁ = aₙ - 5, which means 'subtract 5 from the previous term'). Let's find the first four terms step by step using the recursive definition. We start with a₁ = 2. Then we apply the rule aₙ₊₁ = aₙ - 5: a₂ = a₁ - 5 = 2 - 5 = -3, then a₃ = a₂ - 5 = -3 - 5 = -8, and finally a₄ = a₃ - 5 = -8 - 5 = -13. So the first four terms are 2, -3, -8, -13. 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 2, -3, -8, -13. Great work following the pattern! Choice B adds 5 instead of subtracting 5. 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.
A sequence is defined recursively by a1=3 and an+1=an+4 for n=1,2,3,…. What are the first four terms a1,a2,a3,a4?
Explanation: 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 sequence is defined recursively for n=1,2,3,… by a1=10 and an+1=an−2. What are the first four terms a1,a2,a3,a4?
Explanation: 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 sequence is defined by the explicit formula an=2n+1 for n=1,2,3,…. What is a6?
Explanation: 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 = 2n + 1 lets us find any term directly by plugging in the value of n. To find a₆, we simply substitute n = 6 into the formula: a₆ = 2(6) + 1 = 12 + 1 = 13. So the 6th term is 13. Choice B is correct because it properly evaluates the explicit formula with n = 6 to get 13. Great work following the pattern! Choice A gives 11, which would be a₅ (since 2(5) + 1 = 11), not a₆. When using explicit formulas, make sure you're substituting the correct value of n! The beauty of explicit formulas is that you can jump directly to any term—want the 100th term? Just calculate a₁₀₀ = 2(100) + 1 = 201. No need to find all the terms before it! This is much faster than recursive definitions when you need a term far along in the sequence.
The sequence 3,7,11,15,… is a function with domain n=1,2,3,… where an is the nth term. Which recursive definition generates this sequence?
Explanation: 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 3, 7, 11, 15, ..., let's find the pattern between consecutive terms: from 3 to 7, we add 4. From 7 to 11, we add 4. From 11 to 15, we add 4. This pattern holds for all consecutive terms, so the recursive rule is aₙ₊₁ = aₙ + 4, with starting value a₁ = 3. Choice B is correct because it properly identifies the pattern as adding 4 each time and starts with the correct initial value of 3. Great work following the pattern! Choice A makes an error by saying we add 3 instead of 4: if we started at 3 and added 3, we'd get 3, 6, 9, 12, ... which doesn't match our sequence. Look carefully at the differences between consecutive terms! 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!