This question tests your ability to write arithmetic and geometric sequences in both recursive form (each term from previous) and explicit form (any term directly from its position), and to translate between these forms. Translating between forms: if you have recursive (like a₁ = 3, aₙ₊₁ = aₙ + 5), extract a₁ = 3 and d = 5 (the amount added), then write explicit aₙ = 3 + 5(n - 1). If you have explicit (like aₙ = 2·3^(n-1)), extract a₁ = 2 (when n = 1) and r = 3 (the base), then write recursive a₁ = 2, aₙ₊₁ = 3aₙ. The parameters connect the two forms! Starting from the recursive a₁=6, aₙ₊₁=(1/2)aₙ, this is geometric with r=1/2, so the explicit form is $aₙ=6·(1/2)^{n-1}$ by applying the geometric formula. Choice B correctly translates to the explicit formula for this geometric sequence. A distractor like choice A might use n instead of n-1 in the exponent, but plug in n=1 to check: it should give a₁=6, which $aₙ=6·(1/2)^{n-1}$ does perfectly. Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that d) AND ratios of consecutive terms (constant → geometric with that r). For 5, 8, 11, 14: differences are 3, 3, 3 (arithmetic!), ratios are 8/5, 11/8, 14/11 (not constant). For 3, 6, 12, 24: differences are 3, 6, 12 (not constant), ratios are 2, 2, 2 (geometric!). This two-part check identifies the type reliably. Formula-writing checklist: (1) Identify type (arithmetic or geometric?), (2) Find first term a₁ (just look at the sequence), (3) Find d (subtract consecutive terms) or r (divide consecutive terms), (4) For recursive: state a₁ and write aₙ₊₁ = aₙ + d or aₙ₊₁ = r·aₙ, (5) For explicit: use aₙ = a₁ + (n-1)d or aₙ = a₁·r^(n-1). Follow these steps methodically and you'll get both forms correctly every time!

This question tests your ability to write arithmetic and geometric sequences in both recursive form (each term from previous) and explicit form (any term directly from its position), and to translate between these forms. Arithmetic sequences have constant difference d (add same amount each step): recursive form a₁ = [value], aₙ₊₁ = aₙ + d shows the stepping pattern, while explicit form aₙ = a₁ + (n - 1)d lets you jump directly to any term. Geometric sequences have constant ratio r (multiply by same factor): recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the multiplying pattern, while explicit form aₙ = a₁·r^(n-1) gives direct calculation. Each form has advantages! For depositing $40 each week with a₁=40 after 1 week, it's arithmetic with d=40, so recursive a₁=40, aₙ₊₁=aₙ+40 and explicit aₙ=40+40(n-1). Choice C correctly writes both formulas for this arithmetic sequence. A distractor like choice A might confuse it with geometric growth, but since it's fixed additions, not multiplications, check for constant differences to confirm. Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that d) AND ratios of consecutive terms (constant → geometric with that r). For 5, 8, 11, 14: differences are 3, 3, 3 (arithmetic!), ratios are 8/5, 11/8, 14/11 (not constant). For 3, 6, 12, 24: differences are 3, 6, 12 (not constant), ratios are 2, 2, 2 (geometric!). This two-part check identifies the type reliably. Formula-writing checklist: (1) Identify type (arithmetic or geometric?), (2) Find first term a₁ (just look at the sequence), (3) Find d (subtract consecutive terms) or r (divide consecutive terms), (4) For recursive: state a₁ and write aₙ₊₁ = aₙ + d or aₙ₊₁ = r·aₙ, (5) For explicit: use aₙ = a₁ + (n-1)d or aₙ = a₁·r^(n-1). Follow these steps methodically and you'll get both forms correctly every time!

This question tests your ability to write arithmetic and geometric sequences in both recursive form (each term from previous) and explicit form (any term directly from its position), and to translate between these forms. Translating between forms: if you have recursive (like a₁ = 3, aₙ₊₁ = aₙ + 5), extract a₁ = 3 and d = 5 (the amount added), then write explicit aₙ = 3 + 5(n - 1). If you have explicit (like aₙ = 2·3^(n-1)), extract a₁ = 2 (when n = 1) and r = 3 (the base), then write recursive a₁ = 2, aₙ₊₁ = 3aₙ. The parameters connect the two forms! For the explicit $aₙ=3·2^{n-1}$, evaluate at n=1 to get a₁=3, and recognize r=2 from the base, so recursive is a₁=3, aₙ₊₁=2aₙ. Choice C correctly translates to the recursive definition for this geometric sequence. A distractor like choice A might swap a₁, but always verify by computing the first few terms from the explicit formula to match. Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that d) AND ratios of consecutive terms (constant → geometric with that r). For 5, 8, 11, 14: differences are 3, 3, 3 (arithmetic!), ratios are 8/5, 11/8, 14/11 (not constant). For 3, 6, 12, 24: differences are 3, 6, 12 (not constant), ratios are 2, 2, 2 (geometric!). This two-part check identifies the type reliably. Formula-writing checklist: (1) Identify type (arithmetic or geometric?), (2) Find first term a₁ (just look at the sequence), (3) Find d (subtract consecutive terms) or r (divide consecutive terms), (4) For recursive: state a₁ and write aₙ₊₁ = aₙ + d or aₙ₊₁ = r·aₙ, (5) For explicit: use aₙ = a₁ + (n-1)d or aₙ = a₁·r^(n-1). Follow these steps methodically and you'll get both forms correctly every time!

This question tests your ability to write arithmetic and geometric sequences in both recursive form (each term from previous) and explicit form (any term directly from its position), and to translate between these forms. Arithmetic sequences have constant difference d (add same amount each step): recursive form a₁ = [value], aₙ₊₁ = aₙ + d shows the stepping pattern, while explicit form aₙ = a₁ + (n - 1)d lets you jump directly to any term. Geometric sequences have constant ratio r (multiply by same factor): recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the multiplying pattern, while explicit form aₙ = a₁·r^(n-1) gives direct calculation. Each form has advantages! For this car depreciating 10% per year from $18,000, with a₁ after 1 year as 18,000×0.9=16,200, it's geometric with r=0.9, so recursive a₁=16,200, aₙ₊₁=0.9aₙ and explicit aₙ=16,200·(0.9)^{n-1}. Choice B correctly writes both formulas, ensuring a₁ matches the value after the first year. A distractor like choice A might start with the initial value as a₁, but pay close attention to the problem's definition of a₁ to set it correctly. Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that d) AND ratios of consecutive terms (constant → geometric with that r). For 5, 8, 11, 14: differences are 3, 3, 3 (arithmetic!), ratios are 8/5, 11/8, 14/11 (not constant). For 3, 6, 12, 24: differences are 3, 6, 12 (not constant), ratios are 2, 2, 2 (geometric!). This two-part check identifies the type reliably. Formula-writing checklist: (1) Identify type (arithmetic or geometric?), (2) Find first term a₁ (just look at the sequence), (3) Find d (subtract consecutive terms) or r (divide consecutive terms), (4) For recursive: state a₁ and write aₙ₊₁ = aₙ + d or aₙ₊₁ = r·aₙ, (5) For explicit: use aₙ = a₁ + (n-1)d or aₙ = a₁·r^(n-1). Follow these steps methodically and you'll get both forms correctly every time!

This question tests your ability to write arithmetic and geometric sequences in both recursive form (each term from previous) and explicit form (any term directly from its position), and to translate between these forms. Translating between forms: if you have recursive (like a₁ = 3, aₙ₊₁ = aₙ + 5), extract a₁ = 3 and d = 5 (the amount added), then write explicit aₙ = 3 + 5(n - 1). If you have explicit (like aₙ = 2·3^(n-1)), extract a₁ = 2 (when n = 1) and r = 3 (the base), then write recursive a₁ = 2, aₙ₊₁ = 3aₙ. The parameters connect the two forms! Starting from recursive a₁=-2, aₙ₊₁=aₙ+7, this is arithmetic with d=7, so explicit is aₙ=-2+7(n-1) using the formula. Choice B correctly translates to the explicit formula for this arithmetic sequence. A distractor like choice A might use n instead of (n-1), but test with n=1 (should be -2) and n=2 (should be 5) to see it fits only with (n-1). Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that d) AND ratios of consecutive terms (constant → geometric with that r). For 5, 8, 11, 14: differences are 3, 3, 3 (arithmetic!), ratios are 8/5, 11/8, 14/11 (not constant). For 3, 6, 12, 24: differences are 3, 6, 12 (not constant), ratios are 2, 2, 2 (geometric!). This two-part check identifies the type reliably. Formula-writing checklist: (1) Identify type (arithmetic or geometric?), (2) Find first term a₁ (just look at the sequence), (3) Find d (subtract consecutive terms) or r (divide consecutive terms), (4) For recursive: state a₁ and write aₙ₊₁ = aₙ + d or aₙ₊₁ = r·aₙ, (5) For explicit: use aₙ = a₁ + (n-1)d or aₙ = a₁·r^(n-1). Follow these steps methodically and you'll get both forms correctly every time!

This question tests your ability to write arithmetic and geometric sequences in both recursive form (each term from previous) and explicit form (any term directly from its position), and to translate between these forms. Arithmetic sequences have constant difference $d$ (add same amount each step): recursive form $a_1 = \text{[value]}, a_{n+1} = a_n + d$ shows the stepping pattern, while explicit form $a_n = a_1 + (n - 1)d$ lets you jump directly to any term. Geometric sequences have constant ratio $r$ (multiply by same factor): recursive form $a_1 = \text{[value]}, a_{n+1} = r \cdot a_n$ shows the multiplying pattern, while explicit form $a_n = a_1 \cdot r^{n-1}$ gives direct calculation. Each form has advantages! For the sequence 80, 64, 51.2, 40.96, ..., the ratios are 0.8 each time, so it's geometric with $a_1=80$ and $r=0.8$, leading to recursive $a_1=80, a_{n+1}=0.8a_n$ and explicit $a_n=80 \cdot(0.8)^{n-1}$. Choice B correctly writes both formulas for this geometric sequence. A distractor like choice A might treat it as arithmetic by subtracting differences, but verify ratios are constant while differences aren't to confirm it's geometric. Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that $d$) AND ratios of consecutive terms (constant → geometric with that $r$). For 5, 8, 11, 14: differences are 3, 3, 3 (arithmetic!), ratios are $8/5, 11/8, 14/11$ (not constant). For 3, 6, 12, 24: differences are 3, 6, 12 (not constant), ratios are 2, 2, 2 (geometric!). This two-part check identifies the type reliably. Formula-writing checklist: (1) Identify type (arithmetic or geometric?), (2) Find first term $a_1$ (just look at the sequence), (3) Find $d$ (subtract consecutive terms) or $r$ (divide consecutive terms), (4) For recursive: state $a_1$ and write $a_{n+1} = a_n + d$ or $a_{n+1} = r \cdot a_n$, (5) For explicit: use $a_n = a_1 + (n-1)d$ or $a_n = a_1 \cdot r^{n-1}$. Follow these steps methodically and you'll get both forms correctly every time!

This question tests your ability to write arithmetic and geometric sequences in both recursive form (each term from previous) and explicit form (any term directly from its position), and to translate between these forms. Arithmetic sequences have constant difference d (add same amount each step): recursive form a₁ = [value], aₙ₊₁ = aₙ + d shows the stepping pattern, while explicit form aₙ = a₁ + (n - 1)d lets you jump directly to any term. Geometric sequences have constant ratio r (multiply by same factor): recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the multiplying pattern, while explicit form aₙ = a₁·r^(n-1) gives direct calculation. Each form has advantages! For the sequence 5, 15, 45, 135, ..., the ratios are 3 each time, so it's geometric with a₁=5 and r=3, leading to recursive a₁=5, aₙ₊₁=3aₙ and explicit $aₙ=5·3^{n-1}$. Choice B correctly writes both formulas for this geometric sequence. A distractor like choice A might mistake it for arithmetic, but since ratios are constant and differences increase, it's geometric—great job checking both! Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that d) AND ratios of consecutive terms (constant → geometric with that r). For 5, 8, 11, 14: differences are 3, 3, 3 (arithmetic!), ratios are 8/5, 11/8, 14/11 (not constant). For 3, 6, 12, 24: differences are 3, 6, 12 (not constant), ratios are 2, 2, 2 (geometric!). This two-part check identifies the type reliably. Formula-writing checklist: (1) Identify type (arithmetic or geometric?), (2) Find first term a₁ (just look at the sequence), (3) Find d (subtract consecutive terms) or r (divide consecutive terms), (4) For recursive: state a₁ and write aₙ₊₁ = aₙ + d or aₙ₊₁ = r·aₙ, (5) For explicit: use aₙ = a₁ + (n-1)d or aₙ = a₁·r^(n-1). Follow these steps methodically and you'll get both forms correctly every time!

This question tests your ability to write arithmetic and geometric sequences in both recursive form (each term from previous) and explicit form (any term directly from its position), and to translate between these forms. Arithmetic sequences have constant difference d (add same amount each step): recursive form a₁ = [value], aₙ₊₁ = aₙ + d shows the stepping pattern, while explicit form aₙ = a₁ + (n - 1)d lets you jump directly to any term. Geometric sequences have constant ratio r (multiply by same factor): recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the multiplying pattern, while explicit form aₙ = a₁·r^(n-1) gives direct calculation. Each form has advantages! For this population growth at 3% per year starting from 50,000, with a₁ after 1 year as 50,000×1.03=51,500, it's geometric with r=1.03, so recursive a₁=51,500, aₙ₊₁=1.03aₙ and explicit $aₙ=51,500·(1.03)^{n-1}$. Choice B correctly writes both formulas, aligning a₁ with the value after the first year. A distractor like choice C might use the initial population as a₁, but carefully read the problem to identify what a₁ represents—here it's after 1 year, so adjust accordingly. Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that d) AND ratios of consecutive terms (constant → geometric with that r). For 5, 8, 11, 14: differences are 3, 3, 3 (arithmetic!), ratios are 8/5, 11/8, 14/11 (not constant). For 3, 6, 12, 24: differences are 3, 6, 12 (not constant), ratios are 2, 2, 2 (geometric!). This two-part check identifies the type reliably. Formula-writing checklist: (1) Identify type (arithmetic or geometric?), (2) Find first term a₁ (just look at the sequence), (3) Find d (subtract consecutive terms) or r (divide consecutive terms), (4) For recursive: state a₁ and write aₙ₊₁ = aₙ + d or aₙ₊₁ = r·aₙ, (5) For explicit: use aₙ = a₁ + (n-1)d or aₙ = a₁·r^(n-1). Follow these steps methodically and you'll get both forms correctly every time!

This question tests your ability to write arithmetic and geometric sequences in both recursive form (each term from previous) and explicit form (any term directly from its position), and to translate between these forms. Arithmetic sequences have constant difference d (add same amount each step): recursive form a₁ = [value], aₙ₊₁ = aₙ + d shows the stepping pattern, while explicit form aₙ = a₁ + (n - 1)d lets you jump directly to any term. Geometric sequences have constant ratio r (multiply by same factor): recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the multiplying pattern, while explicit form aₙ = a₁·r^(n-1) gives direct calculation. Each form has advantages! For this sequence 12, 7, 2, -3, ..., the differences are -5 each time, so it's arithmetic with a₁=12 and d=-5, leading to recursive a₁=12, aₙ₊₁=aₙ-5 and explicit aₙ=12-5(n-1). Choice A correctly writes both formulas for this arithmetic sequence. A common distractor like choice B might flip the sign of d, but remember to check the actual differences between terms to confirm if it's increasing or decreasing. Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that d) AND ratios of consecutive terms (constant → geometric with that r). For 5, 8, 11, 14: differences are 3, 3, 3 (arithmetic!), ratios are 8/5, 11/8, 14/11 (not constant). For 3, 6, 12, 24: differences are 3, 6, 12 (not constant), ratios are 2, 2, 2 (geometric!). This two-part check identifies the type reliably. Formula-writing checklist: (1) Identify type (arithmetic or geometric?), (2) Find first term a₁ (just look at the sequence), (3) Find d (subtract consecutive terms) or r (divide consecutive terms), (4) For recursive: state a₁ and write aₙ₊₁ = aₙ + d or aₙ₊₁ = r·aₙ, (5) For explicit: use aₙ = a₁ + (n-1)d or aₙ = a₁·r^(n-1). Follow these steps methodically and you'll get both forms correctly every time!

This question tests your ability to write arithmetic and geometric sequences in both recursive form (each term from previous) and explicit form (any term directly from its position), and to translate between these forms. Translating between forms: if you have recursive (like a₁ = 3, aₙ₊₁ = aₙ + 5), extract a₁ = 3 and d = 5 (the amount added), then write explicit aₙ = 3 + 5(n - 1). If you have explicit (like aₙ = 2·3^(n-1)), extract a₁ = 2 (when n = 1) and r = 3 (the base), then write recursive a₁ = 2, aₙ₊₁ = 3aₙ. The parameters connect the two forms! For the explicit aₙ=9+4(n-1), simplify to see it's arithmetic: a₁=9 (at n=1), and d=4 (coefficient of n term), so recursive is a₁=9, aₙ₊₁=aₙ+4. Choice C correctly translates to the recursive definition for this arithmetic sequence. A distractor like choice A might misidentify it as geometric, but check if differences are constant (yes) versus ratios (no) to confirm the type. Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that d) AND ratios of consecutive terms (constant → geometric with that r). For 5, 8, 11, 14: differences are 3, 3, 3 (arithmetic!), ratios are 8/5, 11/8, 14/11 (not constant). For 3, 6, 12, 24: differences are 3, 6, 12 (not constant), ratios are 2, 2, 2 (geometric!). This two-part check identifies the type reliably. Formula-writing checklist: (1) Identify type (arithmetic or geometric?), (2) Find first term a₁ (just look at the sequence), (3) Find d (subtract consecutive terms) or r (divide consecutive terms), (4) For recursive: state a₁ and write aₙ₊₁ = aₙ + d or aₙ₊₁ = r·aₙ, (5) For explicit: use aₙ = a₁ + (n-1)d or aₙ = a₁·r^(n-1). Follow these steps methodically and you'll get both forms correctly every time!