Arithmetic and Geometric Sequences as Functions - Algebra 1

The first four terms of a sequence are 1, 4, 16, 64. A student claims this can be modeled by both $a_n = 4^{n-1}$ and the recursive formula $a_1 = 1$ , $a_n = 4a_{n-1}$ . Is the student correct, and what is $a_7$ ?
Answer: Student is correct; both formulas are equivalent; $a_7 = 16384$ Checking the sequence 1, 4, 16, 64: each term is 4 times the previous term (geometric with ratio 4). Using $a_n = 4^{n-1}$ : $a_1 = 4^0 = 1$ , $a_2 = 4^1 = 4$ , $a_3 = 4^2 = 16$ , $a_4 = 4^3 = 64$ ✓. The recursive formula $a_1 = 1$ , $a_n = 4a_{n-1}$ also gives the same sequence. So $a_7 = 4^{7-1} = 4^6 = 16384$ . Choice A incorrectly suggests an arithmetic sequence. Choice B has the wrong calculation for $a_7$ ( $4^5 = 1024$ , not $4^6$ ). Choice C proposes an incorrect recursive formula.
Consider the sequence: 12, 8, 4, 0, -4, ... If $a_n$ represents the $n$ th term where $a_1 = 12$ , which recursive formula correctly describes this sequence, and what type of sequence is it?
Answer: $a_n = a_{n-1} - 4$ ; arithmetic sequence with common difference -4 Looking at consecutive terms: $8 - 12 = -4$ , $4 - 8 = -4$ , $0 - 4 = -4$ , $-4 - 0 = -4$ . Since there's a constant difference of -4 between consecutive terms, this is an arithmetic sequence. The recursive formula is $a_n = a_{n-1} + (-4) = a_{n-1} - 4$ . Choice B incorrectly identifies it as geometric ( $12 \times \frac{2}{3} = 8$ , but $8 \times \frac{2}{3} = \frac{16}{3} \neq 4$ ). Choice C has the wrong sign for the common difference. Choice D also incorrectly identifies it as geometric.
Two sequences are given: Sequence X has the explicit formula $x_n = 15 - 2n$ and Sequence Y is defined recursively as $y_1 = 13$ and $y_n = y_{n-1} - 2$ . How do these sequences compare?
Answer: They are identical sequences with the same terms in the same order When comparing sequences, you need to determine whether they're arithmetic or geometric, then examine their specific terms and patterns. Let's start with Sequence X: $x_n = 15 - 2n$ . This is an explicit formula where you can find any term directly. For the first few terms: $x_1 = 15 - 2(1) = 13$ , $x_2 = 15 - 2(2) = 11$ , $x_3 = 15 - 2(3) = 9$ . The sequence is 13, 11, 9, 7, ... with a common difference of -2, making it arithmetic. Now for Sequence Y: Starting with $y_1 = 13$ and the recursive rule $y_n = y_{n-1} - 2$ , we get: $y_1 = 13$ , $y_2 = 13 - 2 = 11$ , $y_3 = 11 - 2 = 9$ . This gives us 13, 11, 9, 7, ... which is also arithmetic with a common difference of -2. Both sequences produce exactly the same terms: 13, 11, 9, 7, 5, ... This makes answer choice D correct. Looking at the wrong answers: A is incorrect because both sequences are arithmetic, not one arithmetic and one geometric. B is wrong because they have the same starting term (13) and the same common difference (-2). C is false because while both are arithmetic, they have identical common differences of -2, not different ones. Study tip: When comparing sequences, always calculate the first few terms of each sequence explicitly. This reveals patterns and relationships that might not be obvious from the formulas alone, especially when one is given explicitly and another recursively.
Write both a recursive formula and an explicit formula for the arithmetic sequence $4, 9, 14, 19, \dots$ .
Answer: Recursive: $a_1=4,\ a_{n+1}=a_n+5$ ; Explicit: $a_n=4+5(n-1)$ This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Arithmetic sequences have a constant difference between consecutive terms: you add (or subtract) the same amount each time. The explicit formula aₙ = a₁ + (n - 1)d uses the first term a₁ and the common difference d to find any term directly, while the recursive form a₁ = [value], aₙ₊₁ = aₙ + d describes how to get each term from the one before. Looking at the sequence 4, 9, 14, 19, ..., we find the common difference by subtracting consecutive terms: 9 - 4 = 5, 14 - 9 = 5, so d = 5. With first term a₁ = 4, the explicit formula is aₙ = 4 + (n - 1)·5. The recursive form is: a₁ = 4, aₙ₊₁ = aₙ + 5. Both formulas describe the pattern 'start at 4, add 5 each time'! Choice B is correct because it properly identifies the first term as a₁ = 4 and the common difference as d = 5, giving the recursive formula a₁ = 4, aₙ₊₁ = aₙ + 5 and explicit formula aₙ = 4 + 5(n-1). Great work! Choice A confuses arithmetic with geometric: this sequence adds the same amount each time, so it's arithmetic, not geometric. The pattern is 'add 5', not 'multiply by 5'. To identify arithmetic vs geometric: subtract consecutive terms (if differences are constant, it's arithmetic with d = that difference), or divide consecutive terms (if ratios are constant, it's geometric with r = that ratio). If neither pattern works, it's neither arithmetic nor geometric!
You deposit money into a jar each week. In week 1 you deposit $10, and each week after that you deposit $5 more than the previous week. Write both a recursive and an explicit formula for the amount deposited in week $n$ (not the total saved).
Answer: Recursive: $a_1=10,\ a_{n+1}=a_n+5$ ; Explicit: $a_n=10+5(n-1)$ This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Arithmetic sequences have a constant difference between consecutive terms: you add (or subtract) the same amount each time. The explicit formula aₙ = a₁ + (n - 1)d uses the first term a₁ and the common difference d to find any term directly, while the recursive form a₁ = [value], aₙ₊₁ = aₙ + d describes how to get each term from the one before. Looking at the deposits: week 1=10, week 2=15 (10+5), week 3=20 (15+5), we find the common difference d=5. With first term a₁=10, the explicit formula is aₙ=10 + (n-1)·5. The recursive form is: a₁=10, aₙ₊₁=aₙ+5. Both formulas describe the pattern 'start at 10, add 5 each time'! Choice A is correct because it properly identifies the first term as a₁=10 and the common difference as d=5, giving the recursive a₁=10, aₙ₊₁=aₙ+5 and explicit aₙ=10+5(n-1). Great work! Choice B confuses arithmetic with geometric: this sequence adds the same amount each time, so it's arithmetic, not geometric. The pattern is adding 5, not multiplying by 5. Real-world clue: if the context mentions 'constant increase' or 'same amount each time,' think arithmetic. If it mentions 'percent growth,' 'doubles each,' or 'multiplies by,' think geometric. The language tells you which type! For recursive formulas, you always need two pieces: (1) the initial value (like a₁ = 5), and (2) the rule for getting the next term (like aₙ₊₁ = aₙ + 3 for arithmetic, or aₙ₊₁ = 2·aₙ for geometric). Without both pieces, the sequence isn't fully defined!
What is the common difference $d$ of the arithmetic sequence $15, 11, 7, 3, \dots$ ?
Answer: $d = -4$ This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Arithmetic sequences have a constant difference between consecutive terms: you add (or subtract) the same amount each time. The explicit formula aₙ = a₁ + (n - 1)d uses the first term a₁ and the common difference d to find any term directly, while the recursive form a₁ = [value], aₙ₊₁ = aₙ + d describes how to get each term from the one before. Looking at the sequence 15, 11, 7, 3, we find the common difference by subtracting consecutive terms: 11 - 15 = -4, 7 - 11 = -4, so d = -4. With first term a₁ = 15, the explicit formula is aₙ = 15 + (n - 1)·(-4). The recursive form is: a₁ = 15, aₙ₊₁ = aₙ + (-4). Both formulas describe the pattern 'start at 15, add -4 each time'! Choice B is correct because it properly identifies the common difference as d = -4. Great work! Choice A has the right idea but gets the common difference wrong: looking at consecutive terms, we see 11-15=-4, not +4; check by comparing a few consecutive terms! To identify arithmetic vs geometric: subtract consecutive terms (if differences are constant, it's arithmetic with d = that difference), or divide consecutive terms (if ratios are constant, it's geometric with r = that ratio). If neither pattern works, it's neither! For recursive formulas, you always need two pieces: (1) the initial value (like a₁ = 5), and (2) the rule for getting the next term (like aₙ₊₁ = aₙ + 3 for arithmetic, or aₙ₊₁ = 2·aₙ for geometric). Without both pieces, the sequence isn't fully defined!
What is the common ratio of the geometric sequence $10,\ 5,\ 2.5,\ 1.25,\ \dots$ ?
Answer: $r=\dfrac{1}{2}$ This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Geometric sequences have a constant ratio between consecutive terms: you multiply by the same factor each time. The explicit formula aₙ = a₁ · $r^{n-1}$ uses the first term a₁ and common ratio r, while the recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the 'multiply to get next term' pattern. To identify this as geometric and find the common ratio, we check the ratio between consecutive terms: 5/10 = 1/2, 2.5/5 = 1/2, 1.25/2.5 = 1/2, confirming common ratio r = 1/2. Each term is half of the previous term, so we multiply by 1/2 to get from one term to the next. Choice A is correct because it properly identifies the common ratio as r = 1/2, which means each term is half the previous term. Great work! Choice B has the reciprocal of the correct ratio: we divide by 2 (multiply by 1/2) to get from one term to the next, not multiply by 2. Check by comparing consecutive terms: 10 × (1/2) = 5, not 10 × 2 = 20! To identify arithmetic vs geometric: subtract consecutive terms (if differences are constant, it's arithmetic with d = that difference), or divide consecutive terms (if ratios are constant, it's geometric with r = that ratio). If neither pattern works, it's neither arithmetic nor geometric!
Which function best models the number of bacteria $B(t)$ after $t$ hours, and how many complete doubling periods will have occurred after 10 hours?
Answer: $B(t) = 200 \cdot 2^{t/3}$ ; 3 complete periods This is a geometric sequence where the bacteria doubles every 3 hours. The initial amount is 200, and after every 3 hours the population is multiplied by 2. So after $t$ hours, there have been $t/3$ doubling periods, giving $B(t) = 200 \cdot 2^{t/3}$ . After 10 hours, there have been $10/3 = 3.33...$ hours worth of doubling, so 3 complete doubling periods. Choice B incorrectly uses $2^{3t}$ . Choice C uses addition instead of multiplication. Choice D uses the wrong base and exponent.
The sequence is defined recursively by $a_1 = 2$ and $a_{n+1} = a_n + 4$ . What is the explicit formula for $a_n$ ?
Answer: $a_n = 2 + 4(n-1)$ This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Arithmetic sequences have a constant difference between consecutive terms: you add (or subtract) the same amount each time. The explicit formula aₙ = a₁ + (n - 1)d uses the first term a₁ and the common difference d to find any term directly, while the recursive form a₁ = [value], aₙ₊₁ = aₙ + d describes how to get each term from the one before. Given recursive formula type, we identify: the recursive a1=2, $a_{n+1}$ = $a_n$ +4 shows it's arithmetic with a₁=2 and d=4 from the added amount. To convert to explicit formula type, we use the arithmetic explicit structure, plugging in a₁ and d, so $a_n$ = 2 + (n-1)·4. So the target formula is $a_n$ = 2 + 4(n-1). Choice B is correct because it properly identifies the first term as a₁ = 2 and the common difference as d = 4, giving the explicit formula $a_n$ = 2 + 4(n-1). Great work! Choice C confuses arithmetic with geometric: this sequence adds the same amount each time, so it's arithmetic, not geometric. The pattern is adding 4 each time, not multiplying by 4. To convert between forms: from recursive to explicit, identify a₁ and d (or r), then plug into the explicit formula. From explicit to recursive, read off a₁ and find d (coefficient of n after simplifying) or r (the base of the exponent). Writing explicit formulas: for arithmetic, use aₙ = a₁ + (n - 1)d (start with first term, add the difference (n-1) times); for geometric, use aₙ = a₁· $r^{n-1}$ (start with first term, multiply by ratio (n-1) times). The (n-1) appears because the first term already includes one application!
Write the recursive formula for the geometric sequence $6,\ 18,\ 54,\ 162,\ \dots$ .
Answer: $a_1=6,\ a_{n+1}=3a_n$ This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Geometric sequences have a constant ratio between consecutive terms: you multiply by the same factor each time. The explicit formula aₙ = a₁ · $r^{n-1}$ uses the first term a₁ and common ratio r, while the recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the 'multiply to get next term' pattern. To identify this as geometric and write formulas, we check the ratio: 18/6 = 3, 54/18 = 3, confirming common ratio r = 3. With a₁ = 6, the explicit formula is aₙ = 6· $3^{n-1}$ . The recursive version: a₁ = 6, aₙ₊₁ = 3·aₙ. This captures 'start at 6, multiply by 3 each time.' Choice B is correct because it properly identifies the first term as a₁ = 6 and the common ratio as r = 3, giving the recursive formula a₁ = 6, aₙ₊₁ = 3aₙ. Great work! Choice D has the right idea but gets the common ratio wrong: looking at consecutive terms, we see 18/6 = 3, giving r = 3, not 2. Check by comparing a few consecutive terms! For recursive formulas, you always need two pieces: (1) the initial value (like a₁ = 6), and (2) the rule for getting the next term (like aₙ₊₁ = 3·aₙ for geometric). Without both pieces, the sequence isn't fully defined!
Write the explicit formula for the geometric sequence $3, 6, 12, 24, \dots$ .
Answer: $a_n = 3\cdot 2^{n-1}$ This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Geometric sequences have a constant ratio between consecutive terms: you multiply by the same factor each time. The explicit formula aₙ = a₁ · $r^{n-1}$ uses the first term a₁ and common ratio r, while the recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the 'multiply to get next term' pattern. To identify this as geometric and write formulas, we check the ratio: 6/3=2, 12/6=2, 24/12=2, confirming common ratio r=2. With a₁=3, the explicit formula is aₙ=3· $2^{n-1}$ . The recursive version: a₁=3, aₙ₊₁=2·aₙ. This captures 'start at 3, multiply by 2 each time.' Choice C is correct because it properly identifies the first term as a₁=3 and the common ratio as r=2, giving the explicit formula aₙ=3· $2^{n-1}$ . Great work! Choice A confuses arithmetic with geometric: this sequence multiplies by the same amount each time, so it's geometric, not arithmetic. The pattern is multiplying by 2, not adding 3. To identify arithmetic vs geometric: subtract consecutive terms (if differences are constant, it's arithmetic with d = that difference), or divide consecutive terms (if ratios are constant, it's geometric with r = that ratio). If neither pattern works, it's neither! Writing explicit formulas: for arithmetic, use aₙ = a₁ + (n - 1)d (start with first term, add the difference (n-1) times); for geometric, use aₙ = a₁· $r^{n-1}$ (start with first term, multiply by ratio (n-1) times). The (n-1) appears because the first term already includes one application!
Write both the explicit formula and the recursive formula for the arithmetic sequence $4,\ 9,\ 14,\ 19,\ \dots$ .
Answer: Explicit: $a_n = 4 + 5(n-1)$ ; Recursive: $a_1=4,\ a_{n+1}=a_n+5$ This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Arithmetic sequences have a constant difference between consecutive terms: you add (or subtract) the same amount each time. The explicit formula aₙ = a₁ + (n - 1)d uses the first term a₁ and the common difference d to find any term directly, while the recursive form a₁ = [value], aₙ₊₁ = aₙ + d describes how to get each term from the one before. Looking at the sequence 4, 9, 14, 19, ..., we find the common difference by subtracting consecutive terms: 9 - 4 = 5, 14 - 9 = 5, so d = 5. With first term a₁ = 4, the explicit formula is aₙ = 4 + (n - 1)·5. The recursive form is: a₁ = 4, aₙ₊₁ = aₙ + 5. Both formulas describe the pattern 'start at 4, add 5 each time'! Choice B is correct because it properly identifies the first term as a₁ = 4 and the common difference as d = 5, giving the explicit formula aₙ = 4 + 5(n-1) and recursive formula a₁ = 4, aₙ₊₁ = aₙ + 5. Great work! Choice A confuses arithmetic with geometric: this sequence adds the same amount each time, so it's arithmetic, not geometric. The pattern is 'add 5', not 'multiply by 5'. To identify arithmetic vs geometric: subtract consecutive terms (if differences are constant, it's arithmetic with d = that difference), or divide consecutive terms (if ratios are constant, it's geometric with r = that ratio). If neither pattern works, it's neither arithmetic nor geometric!
A recursive formula for a sequence is given as $a_1 = 7$ and $a_n = a_{n-1} + 4$ for $n \geq 2$ . Which explicit formula represents the same sequence and what is the value of $a_{15}$ ?
Answer: $a_n = 7 + 4(n-1)$ ; $a_{15} = 63$ The recursive formula shows this is an arithmetic sequence with first term $a_1 = 7$ and common difference $d = 4$ . The explicit formula for an arithmetic sequence is $a_n = a_1 + d(n-1) = 7 + 4(n-1)$ . Therefore $a_{15} = 7 + 4(15-1) = 7 + 4(14) = 7 + 56 = 63$ . Choice A incorrectly uses $4n$ instead of $4(n-1)$ . Choice B switches the first term and common difference. Choice D also switches the values and uses the wrong formula structure.
Translate the explicit formula $a_n=5\cdot 3^{n-1}$ into a recursive definition.
Answer: $a_1=5,\ a_{n+1}=3a_n$ This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). The difference between recursive and explicit formulas: recursive is like climbing stairs one at a time (you need to know the previous term), while explicit is like taking an elevator directly to any floor (you can find the nth term without finding all the ones before it). Both describe the same sequence, just different approaches! Given the explicit formula $a_n = 5 \cdot 3^{n-1}$ , we identify: this is geometric (has the form $a_1 \cdot r^{n-1}$ ) with first term $a_1 = 5$ and common ratio $r = 3$ . To convert to recursive form, we need: $a_1 = 5$ (the starting value), and $a_{n+1} = 3 \cdot a_n$ (multiply by 3 to get the next term). Choice B is correct because it properly identifies the first term as $a_1 = 5$ and the multiplication rule $a_{n+1} = 3 a_n$ , capturing the 'multiply by 3' pattern from the explicit formula. Great work! Choice A confuses arithmetic with geometric: the explicit formula $5 \cdot 3^{n-1}$ shows multiplication (geometric), not addition (arithmetic). The recursive rule should multiply by 3, not add 3! To convert between forms: from recursive to explicit, identify $a_1$ and d (or r), then plug into the explicit formula. From explicit to recursive, read off $a_1$ and find d (coefficient of n after simplifying) or r (the base of the exponent).
Translate the explicit formula $a_n = 5\cdot 2^{n-1}$ into a recursive definition.
Answer: $a_1=5,\ a_{n+1}=2a_n$ This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). The difference between recursive and explicit formulas: recursive is like climbing stairs one at a time (you need to know the previous term), while explicit is like taking an elevator directly to any floor (you can find the nth term without finding all the ones before it). Both describe the same sequence, just different approaches! Given the explicit formula aₙ = 5· $2^{n-1}$ , we identify: this is geometric (has an exponential term) with first term a₁ = 5· $2^{1-1}$ = 5· $2^0$ = 5·1 = 5 and common ratio r = 2 (the base of the exponent). To convert to recursive form, we need a₁ = 5 and the rule aₙ₊₁ = 2·aₙ. So the recursive definition is a₁ = 5, aₙ₊₁ = 2aₙ. Choice C is correct because it properly identifies the first term as a₁ = 5 and the common ratio as r = 2, giving the recursive formula a₁ = 5, aₙ₊₁ = 2aₙ. Great work! Choice D has the correct first term but makes an error in the recursive part: the recursive rule should just multiply by the constant ratio 2, not by $2^{n-1}$ which would make the ratio change each time. Remember: geometric sequences multiply by the same factor each time! For recursive formulas, you always need two pieces: (1) the initial value (like a₁ = 5), and (2) the rule for getting the next term (like aₙ₊₁ = 2·aₙ for geometric). Without both pieces, the sequence isn't fully defined!
A bacteria culture starts with 200 bacteria and triples each hour. Write both a recursive and an explicit formula for the number of bacteria after $n$ hours, where $n=1$ corresponds to the first hour measurement.
Answer: Recursive: $a_1=200,\ a_{n+1}=3a_n$ ; Explicit: $a_n=200\cdot 3^{n-1}$ This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Geometric sequences have a constant ratio between consecutive terms: you multiply by the same factor each time. The explicit formula aₙ = a₁ · $r^{n-1}$ uses the first term a₁ and common ratio r, while the recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the 'multiply to get next term' pattern. For the bacteria: starts at 200, triples so r=3, after 1 hour: 200*3=600, 2 hours:600*3=1800, confirming geometric. With a₁=200, the explicit formula is aₙ=200· $3^{n-1}$ . The recursive version: a₁=200, aₙ₊₁=3·aₙ. This captures 'start at 200, multiply by 3 each time.' Choice B is correct because it properly identifies the first term as a₁=200 and the common ratio as r=3, giving the recursive a₁=200, aₙ₊₁=3aₙ and explicit aₙ=200· $3^{n-1}$ . Great work! Choice A confuses arithmetic with geometric: this sequence multiplies by the same amount each time, so it's geometric, not arithmetic. The pattern is multiplying by 3, not adding 3. Real-world clue: if the context mentions 'constant increase' or 'same amount each time,' think arithmetic. If it mentions 'percent growth,' 'doubles each,' or 'multiplies by,' think geometric. The language tells you which type! Writing explicit formulas: for arithmetic, use aₙ = a₁ + (n - 1)d (start with first term, add the difference (n-1) times); for geometric, use aₙ = a₁· $r^{n-1}$ (start with first term, multiply by ratio (n-1) times). The (n-1) appears because the first term already includes one application!
What is the explicit formula for the arithmetic sequence with $a_1=10$ and common difference $d=-3$ ?
Answer: $a_n=10-3(n-1)$ This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Arithmetic sequences have a constant difference between consecutive terms: you add (or subtract) the same amount each time. The explicit formula aₙ = a₁ + (n - 1)d uses the first term a₁ and the common difference d to find any term directly, while the recursive form a₁ = [value], aₙ₊₁ = aₙ + d describes how to get each term from the one before. Given a₁ = 10 and d = -3, we apply the arithmetic explicit formula: aₙ = a₁ + (n-1)d = 10 + (n-1)·(-3) = 10 - 3(n-1). This gives us the sequence 10, 7, 4, 1, -2, ... (subtracting 3 each time). Choice A is correct because it properly uses the arithmetic explicit formula aₙ = a₁ + (n-1)d with a₁ = 10 and d = -3, giving aₙ = 10 - 3(n-1). Great work! Choice B uses the wrong formula structure: aₙ = 10 - 3n would give a₁ = 10 - 3(1) = 7, not 10. Remember, we need (n-1) in the formula, not just n! Writing explicit formulas: for arithmetic, use aₙ = a₁ + (n - 1)d (start with first term, add the difference (n-1) times); for geometric, use aₙ = a₁· $r^{n-1}$ (start with first term, multiply by ratio (n-1) times). The (n-1) appears because the first term already includes one application!
What is the common ratio of the geometric sequence $8, -4, 2, -1, \dots$ ?
Answer: $r=-\tfrac{1}{2}$ This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Geometric sequences have a constant ratio between consecutive terms: you multiply by the same factor each time. The explicit formula aₙ = a₁ · $r^{n-1}$ uses the first term a₁ and common ratio r, while the recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the 'multiply to get next term' pattern. To identify this as geometric and write formulas, we check the ratio: -4/8=-1/2, 2/(-4)=-1/2, -1/2= -1/2, confirming common ratio r=-1/2. With a₁=8, the explicit formula is aₙ=8· $(-1/2)^{n-1}$ . The recursive version: a₁=8, aₙ₊₁=(-1/2)·aₙ. This captures 'start at 8, multiply by -1/2 each time.' Choice B is correct because it properly identifies the common ratio as r=-1/2 from consecutive divisions. Great work! Choice A has the right idea but gets the common ratio wrong: looking at consecutive terms, we see -4/8=-1/2, not 2. Check by comparing a few consecutive terms! To identify arithmetic vs geometric: subtract consecutive terms (if differences are constant, it's arithmetic with d = that difference), or divide consecutive terms (if ratios are constant, it's geometric with r = that ratio). If neither pattern works, it's neither!
Is the sequence $9, 3, 1, \tfrac{1}{3}, \dots$ arithmetic, geometric, or neither?
Answer: Geometric This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Geometric sequences have a constant ratio between consecutive terms: you multiply by the same factor each time. The explicit formula aₙ = a₁ · $r^{n-1}$ uses the first term a₁ and common ratio r, while the recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the 'multiply to get next term' pattern. To identify this as geometric and write formulas, we check the ratio: 3/9=1/3, 1/3=1/3, (1/3)/(1)=1/3, confirming common ratio r=1/3. With a₁=9, the explicit formula is aₙ=9· $(1/3)^{n-1}$ . The recursive version: a₁=9, aₙ₊₁=(1/3)·aₙ. This captures 'start at 9, multiply by 1/3 each time.' Choice B is correct because it properly identifies the sequence as geometric with constant ratios, not differences. Great work! Choice A confuses arithmetic with geometric: this sequence multiplies by the same amount each time, so it's geometric, not arithmetic. The pattern is multiplying by 1/3, not subtracting a constant. To identify arithmetic vs geometric: subtract consecutive terms (if differences are constant, it's arithmetic with d = that difference), or divide consecutive terms (if ratios are constant, it's geometric with r = that ratio). If neither pattern works, it's neither!
The explicit formula $f(n) = 3 + 7(n-1)$ represents the cost in dollars to rent a movie for $n$ days. What would be the equivalent recursive formula, and what does the coefficient 7 represent in this context?
Answer: $f(1) = 3$ ; $f(n) = f(n-1) + 7$ ; daily fee after first day From $f(n) = 3 + 7(n-1)$ , when $n=1$ : $f(1) = 3 + 7(0) = 3$ . This is an arithmetic sequence with first term 3 and common difference 7, so $f(n) = f(n-1) + 7$ . In context, the initial cost is $3 for the first day, and each additional day costs $7 more. Choice B has the wrong first term and common difference. Choice C incorrectly calculates $f(1) = 10$ . Choice D treats it as a geometric sequence instead of arithmetic.

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