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!

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!

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!

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!

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!

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!

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!

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 $a_n = a_1 + (n - 1)d$ uses the first term $a_1$ and the common difference $d$ to find any term directly, while the recursive form $a_1 = \text{[value]}$, $a_{n+1} = a_n + d$ describes how to get each term from the one before. Given the starting details with $a_1=2$ and $d=-4$, we identify: this is arithmetic since it adds (negative) constant each time. With first term $a_1=2$, the explicit formula is $a_n=2 + (n-1) \cdot(-4)$. The recursive form would be: $a_1=2$, $a_{n+1}=a_n + (-4)$. Both formulas describe the pattern 'start at 2, add -4 each time'! Choice A is correct because it properly identifies the first term as $a_1=2$ 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, not multiplying by -4. Writing explicit formulas: for arithmetic, use $a_n = a_1 + (n - 1)d$ (start with first term, add the difference (n-1) times); for geometric, use $a_n = a_1 \cdot r^{n-1}$ (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!

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!

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!