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Algebra Help: Sequences As Functions And Recursion

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The sequence $a_n$ is defined recursively for $n=1,2,3,\dots$ by $a_1=3$ and $a_{n+1}=3a_n$ . What is the next term after $3, 9, 27$ ?

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Question 1

The sequence $a_n$ is defined recursively for $n=1,2,3,\dots$ by $a_1=3$ and $a_{n+1}=3a_n$ . What is the next term after $3, 9, 27$ ?

$54$
$9$
$81$ (correct answer)
$30$

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 = 3aₙ, which means 'multiply the previous term by 3'). To find any term, you start at the beginning and apply the rule step by step. The sequence starts with 3, then 9 (33), then 27 (93), so the next term is 273 = 81. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice C is correct because it accurately calculates to get 81 by correctly evaluating each term following the recursive rule. Great work following the pattern! Choice B gets the first few terms right but makes an arithmetic error when calculating the fourth term, computing something like 272=54 instead of 27*3=81. With recursive sequences, one arithmetic slip affects all the terms after it! 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!

Question 2

A sequence is defined recursively by $a_1=2$ and $a_{n+1}=a_n-5$ for $n=1,2,3,\dots$ . What are the first four terms $a_1,a_2,a_3,a_4$ ?

$2,\ -3,\ -8,\ -13$ (correct answer)
$2,\ 7,\ 12,\ 17$
$2,\ -3,\ -8,\ -3$
$-3,\ -8,\ -13,\ -18$

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.

Question 3

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 $f(6)$ ?

$5$
$8$ (correct answer)
$13$
$7$

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.

Question 4

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 $f(4)$ ?

$18$
$54$ (correct answer)
$24$
$81$

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!

Question 5

A function $f$ with domain $n=0,1,2,3,\dots$ is defined recursively by $f(0)=3$ and $f(n+1)=f(n)+5$ . What is $f(3)$ ?

$15$
$23$
$13$
$18$ (correct answer)

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.

Question 6

A function $f$ is defined on the positive integers (so the domain is $n = 1,2,3,\dots$ ) by $f(1)=2$ and $f(n+1)=2f(n)$ for $n\ge 1$ . What is $f(5)$ ?

$10$
$32$ (correct answer)
$64$
$16$

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(5) 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) = 22 = 4, then f(3) = 24 = 8, then f(4) = 28 = 16, and f(5) = 216 = 32. So f(5) = 32. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice C is correct because it accurately calculates to get 32. Choice A gets the first few terms right but makes an arithmetic error when calculating f(5), computing 28=16 instead of continuing to 216=32. With recursive sequences, one arithmetic slip affects all the terms after it! 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.

Question 7

A function $f$ is defined only for integers $n = 1,2,3,\dots$ by $f(1)=2$ and $f(n+1)=2f(n)$ . What is $f(4)$ ?

$8$
$10$
$16$ (correct answer)
$32$

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.

Question 8

The sequence $2,\ 5,\ 8,\ 11,\ \dots$ is a function with domain $n=1,2,3,\dots$ . How is this sequence defined recursively?

$a_1=2,\ a_{n+1}=a_n+3$ (correct answer)
$a_1=5,\ a_{n+1}=a_n+3$
$a_1=2,\ a_{n+1}=a_n\cdot 3$
$a_1=2,\ a_{n+1}=a_n-3$

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, 5, 8, 11, ..., let's find the pattern between consecutive terms: from 2 to 5, we add 3. From 5 to 8, we add 3 again. From 8 to 11, we add 3 once more. This pattern holds for all consecutive terms, so the recursive rule is aₙ₊₁ = aₙ + 3, with starting value a₁ = 2. Choice A is correct because it correctly identifies the pattern as adding 3 each time and starts with the right initial value of 2. Great work following the pattern! Choice B has the wrong starting value, using a₁ = 5 instead of a₁ = 2. 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)? Once you spot the pattern, write it as aₙ₊₁ = [rule using aₙ], and don't forget to state your starting value!

Question 9

A sequence is defined recursively by $a_1=4$ and $a_{n+1}=a_n+3$ for $n=1,2,3,\dots$ . What is $a_5$ ?

$13$
$16$ (correct answer)
$19$
$15$

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'). 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.

Question 10

A sequence is defined recursively by $a_1 = 4$ and $a_{n+1} = a_n + 3$ for $n = 1,2,3,\dots$ . What is $a_5$ ?

$19$
$16$ (correct answer)
$15$
$13$

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 $a_1 = 4$ ) and a rule for finding the next one (like $a_{n+1} = a_n + 3$ , 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 $a_5$ step by step using the recursive definition. We start with $a_1 = 4$ . Then we apply the rule $a_{n+1} = a_n + 3$ : $a_2 = 4 + 3 = 7$ , then $a_3 = 7 + 3 = 10$ , then $a_4 = 10 + 3 = 13$ , and $a_5 = 13 + 3 = 16$ . So $a_5 = 16$ . 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.