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.

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!

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

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

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!

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!

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!

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!

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.

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.