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, 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 mistakenly adding more terms or skipping steps, but remember to build sequentially from the initials. 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! Recursive vs explicit: recursive definitions are often simpler to state ('add the previous two terms') but harder to evaluate for large n (must calculate all terms before it). Explicit formulas are harder to find ('aₙ = formula just in n') but let you jump to any term: a₁₀₀ directly without finding a₁ through a₉₉. For Fibonacci, there IS an explicit formula (Binet's formula involving golden ratio), but it's complex—recursion is actually clearer! 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: 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. Recursive definitions specify how to get each term from previous term(s): for this sequence, a₁ = 2 and $a_{n+1}$ = a_n² - 1, meaning each term is the square of the previous minus one. Unlike linear recursions, this nonlinear one grows quickly, and you must compute step by step. Starting with a₁ = 2, then a₂ = 2² - 1 = 4 - 1 = 3, a₃ = 3² - 1 = 9 - 1 = 8, a₄ = 8² - 1 = 64 - 1 = 63. Choice C correctly evaluates recursively as 63. A distractor like B (48) might result from mistakenly doing 7² - 1 or skipping a step, but always apply the rule precisely to the previous term. Evaluating recursive sequences systematically: (1) Write down the initial value(s) clearly: a₁ = 2, (2) Set up a table or list: n = 1, 2, 3, ... , (3) Apply the recursive rule one step at a time, computing each fully before moving on, (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 can lead to rapid growth in nonlinear cases like this, making large n computationally intensive.

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 $f(1) = 1$, $f(2) = 1$ 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!

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.

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: 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. 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. 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!