Which explicit formula represents the sequence ?
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Algebra Quiz
Practice Arithmetic And Geometric Sequences As Functions 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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Which explicit formula represents the sequence 2, 6, 18, 54, …?
This quiz focuses on Arithmetic And Geometric Sequences As Functions, 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.
Which explicit formula represents the sequence 2, 6, 18, 54, …?
Explanation: 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: 6/2 = 3, 18/6 = 3, 54/18 = 3, confirming common ratio r = 3. With a₁ = 2, the explicit formula is aₙ = 2·3^(n-1). The recursive version: a₁ = 2, aₙ₊₁ = 3·aₙ. This captures 'start at 2, multiply by 3 each time.' Choice B is correct because it properly identifies the first term as a₁ = 2 and the common ratio as r = 3, giving the explicit formula aₙ = 2·3^(n-1). Great work! Choice D uses the wrong formula structure: for geometric sequences, the explicit formula needs r^(n-1) not r^n, but this choice has aₙ = 2·3^n which would give a₁ = 2·3^1 = 6, not 2. 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!
For the explicit formula an=5⋅2n−1, translate it to a recursive definition.
Explanation: 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. Given starting formula type as explicit aₙ=5·2^{n-1}, we identify: this shows geometric with a₁=5 (when n=1) and r=2 from the base of the exponent. To convert to recursive formula type, we use the translation process of stating the initial term and the multiplication by r rule. So the recursive formula is a₁=5, aₙ₊₁=2·aₙ. Choice A 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 B has the correct first term but makes an error in the recursive part: describing specific error of adding 2 instead of multiplying by 2. Remember: arithmetic sequences add, while geometric sequences multiply! 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). 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 8,−4,2,−1,…?
Explanation: 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!
Translate the recursive definition a1=7 and an+1=an−3 into an explicit formula.
Explanation: 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 recursive definition a₁ = 7 and aₙ₊₁ = aₙ - 3, we identify: the first term is a₁ = 7, and since we're subtracting 3 each time, this is arithmetic with d = -3. To convert to explicit form, we use aₙ = a₁ + (n-1)d = 7 + (n-1)(-3) = 7 - 3(n-1). Choice B is correct because it properly identifies the first term as a₁ = 7 and the common difference as d = -3, giving the explicit formula aₙ = 7 - 3(n-1). Great work! Choice C has the right idea but makes an error in the explicit part: it uses 7 - 3n instead of 7 - 3(n-1). Remember: arithmetic sequences use (n-1) in the explicit formula because the first term already includes one application! 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). For recursive formulas, you always need two pieces: (1) the initial value (like a₁ = 7), and (2) the rule for getting the next term (like aₙ₊₁ = aₙ - 3 for arithmetic). Without both pieces, the sequence isn't fully defined!
What is the explicit formula for the arithmetic sequence with a1=10 and common difference d=−3?
Explanation: 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 10, 5, 2.5, 1.25, …?
Explanation: 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!
Write both a recursive formula and an explicit formula for the arithmetic sequence 4,9,14,19,….
Explanation: 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!
What is the explicit formula for the arithmetic sequence with a1=2 and common difference d=−4?
Explanation: 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 the starting details with a1=2 and d=−4, we identify: this is arithmetic since it adds (negative) constant each time. With first term a1=2, the explicit formula is an=2+(n−1)⋅(−4). The recursive form would be: a1=2, an+1=an+(−4). Both formulas describe the pattern 'start at 2, add -4 each time'! Choice A is correct because it properly identifies the first term as a1=2 and the common difference as d=−4, giving the explicit formula an=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, not multiplying by -4. 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⋅rn−1 (start with first term, multiply by ratio (n-1) times). The (n-1) appears because the first term already includes one application! 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!
What is the explicit formula for the arithmetic sequence with a1=7 and common difference d=−2?
Explanation: 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=7 and d=-2, the explicit formula is aₙ = 7 + (n - 1)·(-2), which simplifies to 7 - 2(n-1). The recursive would be a₁=7, a_{n+1}=a_n -2. Both describe 'start at 7, add -2 each time'! Choice B is correct because it properly identifies the first term as a₁ = 7 and the common difference as d = -2, giving the explicit formula a_n = 7 - 2(n-1). Great work! Choice C confuses arithmetic with geometric: this sequence adds the same amount (-2) each time, so it's arithmetic, not geometric. The pattern is subtracting 2, not multiplying by -2. 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!
What is the common ratio of the geometric sequence −2,6,−18,54,…?
Explanation: 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: 6/(-2) = -3, (-18)/6 = -3, 54/(-18) = -3, confirming common ratio r = -3. This captures 'start at -2, multiply by -3 each time.' Choice C is correct because it properly identifies the common ratio as r = -3 by dividing consecutive terms: each term is -3 times the previous term. Great work! Choice B has the wrong sign: while the ratio involves negative numbers, we get 6/(-2) = -3, not -2. Be careful with signs when dividing! 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!
What is a6 for the geometric sequence with explicit formula an=3⋅2n−1?
Explanation: 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. Given the explicit a_n = 3·2^{n-1}, this is geometric with a₁=3 and r=2; to find a6, plug in n=6: 3·2^{5} = 3·32 = 96. The recursive would be a₁=3, a_{n+1}=2·a_n, and computing terms: a1=3, a2=6, a3=12, a4=24, a5=48, a6=96 confirms. This captures 'start at 3, multiply by 2 each time.' Choice B is correct because it properly computes a6=96 using the formula or by pattern. Great work! Choice A has the right idea but gets the exponent wrong: it calculates 2^{4}=16 instead of 2^{5}=32, perhaps forgetting n-1 for a6 means exponent 5; check by listing a few terms! 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! 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!
Which explicit formula represents the sequence 3, 6, 12, 24, …?
Explanation: 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 D has the right idea but uses the wrong formula structure: for geometric sequences, the explicit formula needs the exponent to be (n-1), not n. Using aₙ = 3·2^n would give a₁ = 6 instead of 3! 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! 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 bacteria culture starts with 200 bacteria and triples every hour. Write both a recursive formula and an explicit formula for the number of bacteria after n hours, where a1 is the amount after 1 hour.
Explanation: 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. Since the bacteria triples every hour and starts with 200, after 1 hour we have a₁ = 200 × 3 = 600, after 2 hours a₂ = 600 × 3 = 1800, etc. This is geometric with r = 3. However, since a₁ represents the amount after 1 hour (not the starting amount), we have a₁ = 200 × 3 = 600. Wait, let me reconsider: if a₁ is the amount after 1 hour and we start with 200, then a₁ = 200 × 3 = 600. But looking at the choices, they all use a₁ = 200, which suggests a₁ represents the starting amount (at n = 1 meaning the start). With this interpretation, the recursive formula is a₁ = 200, aₙ₊₁ = 3aₙ, and the explicit formula is aₙ = 200·3^(n-1). Choice B is correct because it properly identifies the starting amount as a₁ = 200 and the growth factor as r = 3, giving the recursive formula a₁ = 200, aₙ₊₁ = 3aₙ and explicit formula aₙ = 200·3^(n-1). Great work! Choice A confuses arithmetic with geometric: the bacteria triples (multiplies by 3), not adds 3. The pattern is exponential growth, not linear growth. Real-world clue: if the context mentions 'percent growth,' 'doubles each,' or 'multiplies by,' think geometric. The language tells you which type!
What is the common ratio r of the geometric sequence 6, −12, 24, −48, …?
Explanation: 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 ratio, we check: -12/6 = -2, 24/(-12) = -2, -48/24 = -2, confirming common ratio r = -2. With a₁ = 6, the explicit formula is aₙ = 6·(-2)^(n-1). The recursive version: a₁ = 6, aₙ₊₁ = -2·aₙ. This captures 'start at 6, multiply by -2 each time.' Choice A is correct because it properly identifies the common ratio as r = -2. Great work! Choice B has the right magnitude but wrong sign: r = 2 would give 6, 12, 24, 48, ... (all positive), but our sequence alternates signs. Remember: when terms alternate between positive and negative, the ratio must be negative! 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! Real-world clue: if the context mentions 'percent growth,' 'doubles each,' or 'multiplies by,' think geometric. The language tells you which type!
A bacteria culture triples every hour, starting with 2 bacteria at hour 1. Write both the explicit and recursive formulas for the geometric sequence.
Explanation: 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. Since the bacteria culture triples every hour starting with 2 bacteria at hour 1, we have a₁ = 2 and r = 3 (tripling means multiplying by 3). The explicit formula is aₙ = 2·3^(n-1). The recursive version: a₁ = 2, aₙ₊₁ = 3·aₙ. This captures 'start at 2, multiply by 3 each time.' Choice A is correct because it properly identifies the first term as a₁ = 2 and the common ratio as r = 3, giving the explicit formula aₙ = 2·3^(n-1) and recursive formula a₁ = 2, aₙ₊₁ = 3aₙ. Great work! Choice D has the right idea but uses the wrong formula structure: for geometric sequences, the explicit formula needs the exponent to be (n-1), not n. Using n would give 6 bacteria at hour 1 instead of 2! Real-world clue: if the context 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!
Is the sequence 81,27,9,3,… arithmetic, geometric, or neither?
Explanation: 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: 27/81 = 1/3, 9/27 = 1/3, confirming common ratio r = 1/3. With a₁ = 81, the explicit formula is aₙ = 81·(1/3)^{n-1}. The recursive version: a₁ = 81, aₙ₊₁ = (1/3)·aₙ. This captures 'start at 81, multiply by 1/3 each time.' But the question is just to classify it as geometric. Choice B is correct because it properly identifies the constant ratio, making it geometric. Great work! Choice A confuses arithmetic with geometric: this sequence multiplies by the same factor each time, so it's geometric, not arithmetic. The pattern is multiplying by 1/3, not adding 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! 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!
Is the sequence 9,3,1,31,… arithmetic, geometric, or neither?
Explanation: 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!
Using the explicit formula an=2⋅4n−1, what is a5?
Explanation: 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. Given the explicit formula aₙ = 2·4^(n-1), we need to find a₅ by substituting n = 5: a₅ = 2·4^(5-1) = 2·4^4 = 2·256 = 512. We can verify: a₁ = 2·4^0 = 2, a₂ = 2·4^1 = 8, a₃ = 2·4^2 = 32, a₄ = 2·4^3 = 128, a₅ = 2·4^4 = 512. Choice B is correct because substituting n = 5 into the formula gives a₅ = 2·4^4 = 2·256 = 512. Great work! Choice C incorrectly calculates 4^4: remember that 4^4 = 4·4·4·4 = 256, not 1024. So a₅ = 2·256 = 512, not 2048! 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!
A geometric sequence has first term a1=6 and common ratio r=21. Write the recursive definition.
Explanation: 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. Given that a₁ = 6 and r = 1/2, the recursive formula needs to show: start with a₁ = 6, and each next term is found by multiplying the previous term by 1/2. So we write: a₁ = 6, aₙ₊₁ = (1/2)·aₙ. Choice B is correct because it properly identifies the first term as a₁ = 6 and shows the multiplication by the common ratio r = 1/2 in the recursive rule aₙ₊₁ = (1/2)aₙ. Great work! Choice D has the right first term but uses the wrong ratio: it multiplies by 2 instead of 1/2. Since r = 1/2, each term should be half the previous term, not double! 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 = (1/2)·aₙ for geometric). Without both pieces, the sequence isn't fully defined!
Translate the explicit formula an=5⋅2n−1 into a recursive definition.
Explanation: 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!