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! 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 the sequence 50, 35, 24.5, 17.15, ..., the common ratio is 0.7, making it geometric with a₁=50 and r=0.7, so recursive a₁=50, aₙ₊₁=0.7aₙ and explicit $aₙ=50·(0.7)^{n-1}$. Choice B correctly identifies the sequence type and formulas, matching the decaying pattern. Options like A treat it as arithmetic subtraction, but ratios confirm it's geometric—you're building strong skills here! 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. 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, 15, 45, 135: differences are 10, 30, 90 (not constant), but ratios are 15/5 = 3, 45/15 = 3, 135/45 = 3 (constant!), so this is geometric with r = 3. With first term a₁ = 5 and ratio r = 3, we get recursive: a₁ = 5, aₙ₊₁ = 3aₙ and explicit: aₙ = 5 · 3^(n-1). Choice C correctly identifies this as a geometric sequence and provides both formulas. Choice D incorrectly treats it as arithmetic with d = 3, which would give the sequence 5, 8, 11, 14,... instead of 5, 15, 45, 135,.... Always check both differences and ratios to determine sequence type. If ratios are constant, it's geometric; if differences are constant, it's arithmetic.

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. Here we start with 40 points in week 1 and add 15 points each week, so a₁ = 40 and d = 15, giving us recursive: a₁ = 40, aₙ₊₁ = aₙ + 15 and explicit: aₙ = 40 + 15(n - 1). Choice B correctly writes both formulas with the right starting value and weekly increase. Choice A incorrectly swaps the values, starting with 15 and adding 40 each week, which doesn't match the problem. Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that d) AND ratios of consecutive terms (constant → geometric with that r). 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).

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 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. Looking at the sequence 12, 7, 2, -3, we calculate differences: 7 - 12 = -5, 2 - 7 = -5, -3 - 2 = -5, confirming this is arithmetic with d = -5. With first term a₁ = 12 and common difference d = -5, the recursive definition is a₁ = 12, aₙ₊₁ = aₙ - 5. Choice A correctly identifies both the starting value and the negative common difference. Choice B incorrectly adds 5 instead of subtracting, which would create the increasing sequence 12, 17, 22, 27,... instead of the given decreasing sequence. When finding the common difference, always subtract: (next term) - (current term). A negative result means you subtract in the recursive formula!

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 battery model starting at 90% and retaining 80% (r=0.8) each hour, it's geometric with a₁=90, so recursive a₁=90, aₙ₊₁=0.8aₙ and explicit $aₙ=90·(0.8)^{n-1}$. Choice B correctly writes both formulas, modeling the exponential decay. Choices like A use arithmetic subtraction, but multiplication by 0.8 makes it geometric—fantastic application to real life! 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. 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. For 10% depreciation, the car retains 90% of its value each year, so we multiply by 0.90. Starting with V₀ = 24000, we get recursive: V₀ = 24000, Vₙ₊₁ = 0.90Vₙ and explicit: Vₙ = 24000 · 0.90ⁿ (note: exponent is n because we start at V₀). Choice B correctly models depreciation with the right decay factor and starting index. Choice D incorrectly uses 1.10, which would represent 10% growth (appreciation) rather than depreciation. For depreciation problems, subtract the percentage from 100%: 100% - 10% = 90% = 0.90. This gives the retention factor, not the loss factor!