Arithmetic and Geometric Sequences as Functions - Algebra 2

The geometric sequence is $50,\ 45,\ 40.5,\ 36.45,\ \dots$ with domain $n \in \{1,2,3,\dots\}$ . Write both a recursive formula and an explicit formula for $a_n$ .
Answer: Recursive: $a_1=50,\ a_{n+1}=0.9a_n$ ; Explicit: $a_n=50\cdot 0.9^{n-1}$ 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. 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 the sequence 50, 45, 40.5, 36.45, ...: ratios are 45/50 = 0.9, 40.5/45 = 0.9, 36.45/40.5 = 0.9 (constant!), so this is geometric with a₁ = 50 and r = 0.9, giving recursive a₁ = 50, aₙ₊₁ = 0.9aₙ and explicit aₙ = 50· $0.9^{n-1}$ . Choice B correctly writes both formulas with a₁ = 50 as the first term and r = 0.9 as the common ratio. Choice A incorrectly treats this as arithmetic (subtracting 5), C has wrong first term 45, and D has the wrong ratio 1.1 (growth instead of decay). Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that d) AND ratios of consecutive terms (constant → geometric with that r). For 50, 45, 40.5, 36.45: differences are -5, -4.5, -4.05 (not constant), but ratios are 0.9, 0.9, 0.9 (constant!) - this two-part check confirms it's geometric with r = 0.9!
Given the explicit formula $a_n=7\cdot 2^{n-1}$ for $n\ge 1$ , write the equivalent recursive definition.
Answer: $a_1=7,\ a_{n+1}=2a_n$ 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 explicit (like aₙ = 7· $2^{n-1}$ ), extract a₁ = 7 (when n = 1: a₁ = $7·2^0$ = 7·1 = 7) and r = 2 (the base), then write recursive a₁ = 7, aₙ₊₁ = 2aₙ. The explicit formula aₙ = 7· $2^{n-1}$ represents a geometric sequence with first term 7 and ratio 2, so the recursive definition is a₁ = 7, aₙ₊₁ = 2aₙ. Choice C correctly identifies both the starting value 7 and the common ratio of 2 (multiplying by 2 each step). Choice A incorrectly treats this as arithmetic (adding 2), B starts with 14 instead of 7, and D also treats it as arithmetic (adding 7). 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· $r^{n-1}$ . The parameters connect the two forms!
Write both recursive and explicit formulas for the geometric sequence $80,\ 64,\ 51.2,\ 40.96,\ \dots$ (domain: positive integers).
Answer: Recursive: $a_1=80,\ a_{n+1}=0.8a_n$ ; Explicit: $a_n=80\cdot(0.8)^{n-1}$ 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!
Translate the explicit formula $a_n=3\cdot 2^{n-1}$ into an equivalent recursive definition (domain: positive integers).
Answer: $a_1=3,\ a_{n+1}=2a_n$ 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!
Write both recursive and explicit formulas for the sequence $5,\ 15,\ 45,\ 135,\ \dots$ (domain: positive integers).
Answer: Recursive: $a_1=5,\ a_{n+1}=3a_n$ ; Explicit: $a_n=5\cdot 3^{n-1}$ 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 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!
The sequence is defined recursively by $a_1=5$ and $a_{n+1}=a_n+8$ for $n\ge 1$ . Write the explicit formula for $a_n$ (domain: positive integers).
Answer: $a_n=5+8(n-1)$ 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. 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! Given recursive a₁=5, aₙ₊₁=aₙ+8, it's arithmetic with d=8, so explicit aₙ=5+8(n-1) by adding d repeatedly from a₁. Choice C correctly translates to the explicit formula, yielding terms like a₂=13, a₃=21. Distractors like A assume geometric, but constant addition confirms arithmetic— you're getting better with each practice! 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!
A sequence has the explicit formula $f(n) = 12 + (n-1)(-3)$ . If this sequence is rewritten in recursive form starting with $f(3)$ , which of the following is correct?
Answer: $f(3) = 6$ ; $f(n) = f(n-1) - 3$ for $n \geq 4$ When you encounter a sequence with an explicit formula that needs to be converted to recursive form, you're working with the relationship between consecutive terms. The key is finding both the starting value and the pattern that connects each term to the next. First, let's find $f(3)$ using the explicit formula $f(n) = 12 + (n-1)(-3)$ . Substituting $n = 3$ : $f(3) = 12 + (3-1)(-3) = 12 + 2(-3) = 12 - 6 = 6$ . To determine the recursive pattern, we need to see how consecutive terms relate. Let's calculate $f(4)$ : $f(4) = 12 + (4-1)(-3) = 12 + 3(-3) = 12 - 9 = 3$ . Since $f(4) = 3$ and $f(3) = 6$ , we see that $f(4) = f(3) - 3$ . This confirms the recursive relationship is $f(n) = f(n-1) - 3$ . Now let's examine why the wrong answers fail. Choice A incorrectly calculates $f(3) = 3$ instead of 6, likely confusing it with $f(4)$ . Choice B has the wrong starting value ( $f(3) = 9$ ) and the wrong recursive pattern (adding 3 instead of subtracting). Choice C correctly identifies that $f(3) = 6$ but incorrectly uses $f(n) = f(n-1) + 3$ , which would create an increasing sequence rather than the decreasing one we actually have. Study tip: When converting explicit to recursive form, always calculate at least two consecutive terms to verify both the starting value and the direction of change. The coefficient of $(n-1)$ in arithmetic sequences tells you the common difference.
A bacteria culture starts with 80 bacteria and increases by 10% each hour. Let $P(n)$ be the population after $n$ hours, with domain the nonnegative integers and $P(0)=80$ . Model this situation with both a recursive and an explicit formula.
Answer: Recursive: $P(0)=80,\ P(n+1)=1.1P(n)$ ; Explicit: $P(n)=80\cdot 1.1^{n}$ 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. Since the bacteria increases by 10% each hour, each population is 1.1 times the previous (100% + 10% = 110% = 1.1), starting with P(0) = 80, giving recursive: P(0) = 80, P(n+1) = 1.1P(n), and explicit: P(n) = 80· $1.1^n$ (note: exponent is n, not n-1, because we start at n = 0). Choice C correctly models the 10% growth with ratio 1.1 and uses the proper exponent n since the domain starts at 0. Choice A incorrectly treats this as arithmetic (adding 10), B uses n-1 in the exponent when starting at n = 1 is incorrect, and D models decay (0.9) instead of growth. Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that d) AND ratios of consecutive terms (constant → geometric with that r). When the domain starts at n = 0, the explicit formula uses $r^n$ ; when it starts at n = 1, use $r^{n-1}$ !
Translate the recursive definition $a_1=48$ and $a_{n+1}=\tfrac{2}{3}a_n$ into an explicit formula for $a_n$ (for $n\ge 1$ ).
Answer: $a_n=48\left(\tfrac{2}{3}\right)^{n-1}$ 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. Translating between forms: if you have recursive (like a₁ = 48, aₙ₊₁ = ⅔aₙ), extract a₁ = 48 and r = ⅔ (the multiplying factor), then write explicit aₙ = 48·(⅔)^(n-1). The recursive definition shows a geometric sequence with first term 48 and ratio ⅔, so the explicit formula is aₙ = 48·(⅔)^(n-1). Choice A correctly applies the geometric sequence explicit formula with the proper exponent (n-1) for a sequence starting at n = 1. Choice B has the wrong exponent (n instead of n-1), while C and D incorrectly treat this as arithmetic by adding ⅔ or 48. 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· $r^{n-1}$ . The parameters connect the two forms!
The sequence is defined recursively by $a_1=6$ and $a_{n+1}=\tfrac{1}{2}a_n$ . Write the explicit formula for $a_n$ (treating it as a function with integer domain $n\ge 1$ ).
Answer: $a_n=6\left(\tfrac{1}{2}\right)^{n-1}$ 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. 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. The recursive definition tells us a₁ = 6 and each term is multiplied by ½ to get the next term (r = ½), so the explicit formula is aₙ = 6·(½)^(n-1). Choice B correctly applies the geometric sequence explicit formula with first term 6 and ratio ½. Choice A has the wrong exponent (n instead of n-1), while C and D incorrectly treat this as arithmetic by adding ½ or 6 instead of multiplying. Translating between forms: if you have recursive (like a₁ = 6, aₙ₊₁ = ½·aₙ), extract a₁ = 6 and r = ½ (the multiplying factor), then write explicit aₙ = 6·(½)^(n-1). Follow these steps methodically and you'll get both forms correctly every time!
A student saves money by depositing $25 each week into a jar and does not remove any money. After the first week, the jar has $25. Let $S_n$ be the total saved after $n$ weeks (integer domain $n\ge 1$ ). Write both a recursive and an explicit formula for $S_n$ .
Answer: Recursive: $S_1=25,\ S_{n+1}=S_n+25$ ; Explicit: $S_n=25+25(n-1)$ 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 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. The student saves $25 each week, starting with $25 after week 1, so this is arithmetic with S₁ = 25 and d = 25, giving recursive: S₁ = 25, Sₙ₊₁ = Sₙ + 25 and explicit: Sₙ = 25 + 25(n - 1) = 25n. Choice A correctly models the savings with both formulas. Choice C incorrectly starts with S₁ = 0, which would mean no money after the first week of saving, contradicting the problem statement. For savings problems, be careful about what happens in week 1: if $25 is deposited in week 1, then S₁ = 25, not S₁ = 0. The explicit formula can be simplified: Sₙ = 25 + 25(n - 1) = 25 + 25n - 25 = 25n.
Write both recursive and explicit formulas for the arithmetic sequence $12,\ 7,\ 2,\ -3,\ \dots$ (treat the sequence as a function with domain the positive integers).
Answer: Recursive: $a_1=12,\ a_{n+1}=a_n-5$ ; Explicit: $a_n=12-5(n-1)$ 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 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!
A student deposits $40 into a jar each week and does not remove any money. Let $a_n$ be the total amount in the jar after $n$ weeks, with $a_1=40$ . Write both recursive and explicit formulas (domain: positive integers).
Answer: Recursive: $a_1=40,\ a_{n+1}=a_n+40$ ; Explicit: $a_n=40+40(n-1)$ 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 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!
A theater charges $15 for admission and plans to increase the price by $2 each month. If $a_n$ represents the admission price in the nth month, which recursive formula correctly models this situation, and what will be the price in the 8th month?
Answer: $a_1 = 15, a_n = a_{n-1} + 2$ for $n \geq 2$ ; Price in 8th month: $29 The initial price is $15, so $a_1 = 15$ . Each month the price increases by $2, so $a_n = a_{n-1} + 2$ . For the 8th month: $a_8 = 15 + 7(2) = 29$ . Choice B incorrectly starts at $17. Choice C uses $a_0$ which would make the 8th month $a_8 = 15 + 8(2) = 31$ . Choice D incorrectly adds $15 each month instead of $2.
For the sequence defined explicitly by $a_n=9\left(\tfrac{1}{2}\right)^{n-1}$ for integers $n\ge 1$ , write the equivalent recursive definition.
Answer: $a_1=9,\ a_{n+1}=\tfrac{1}{2}a_n$ 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 explicit (like aₙ = a₁· $r^{n-1}$ ), extract a₁ (when n = 1) and r (the base), then write recursive a₁ = [value], aₙ₊₁ = r·aₙ. From the explicit formula aₙ = 9 $(1/2)^{n-1}$ , we identify a₁ = 9 (coefficient) and r = 1/2 (base), giving recursive: a₁ = 9, aₙ₊₁ = (1/2)aₙ. Choice B correctly translates from explicit to recursive form with the right starting value and ratio. Choice D incorrectly uses r = 2 instead of r = 1/2, which would create a doubling sequence (9, 18, 36,...) instead of the halving sequence (9, 4.5, 2.25,...). When the explicit formula has a fraction as the base, that same fraction appears as the multiplier in the recursive formula. Don't invert it!
A phone battery is at 90% at the start of the day and then retains 80% of its charge at the end of each hour (due to constant use). Let $a_n$ be the battery percentage after $n-1$ hours (so $a_1=90$ ), with domain $n\in\{1,2,3,\dots\}$ . Write both a recursive definition and an explicit formula.
Answer: Recursive: $a_1=90,\ a_{n+1}=0.8a_n$ ; Explicit: $a_n=90\cdot(0.8)^{n-1}$ 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 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!
What is the recursive definition for the arithmetic sequence $12,\ 7,\ 2,\ -3,\dots$ (integer domain $n\ge 1$ )?
Answer: $a_1=12,\ a_{n+1}=a_n-5$ 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!
A car worth $\$18{,}000$ depreciates by 10% each year. Let $a_n$ be the value after $n$ years, with $a_1$ the value after 1 year. Write both recursive and explicit formulas (domain: positive integers).
Answer: Recursive: $a_1=16{,}200,\ a_{n+1}=0.9a_n$ ; Explicit: $a_n=16{,}200\cdot(0.9)^{n-1}$ 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 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!
The arithmetic sequence is $14,\ 9,\ 4,\ -1,\ \dots$ with domain $n \in \{1,2,3,\dots\}$ . Write both a recursive formula and an explicit formula for $a_n$ .
Answer: Recursive: $a_1=14,\ a_{n+1}=a_n-5$ ; Explicit: $a_n=14-5(n-1)$ 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 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. For the sequence 14, 9, 4, -1, ...: differences are 9-14 = -5, 4-9 = -5, -1-4 = -5 (constant!), so this is arithmetic with a₁ = 14 and d = -5, giving recursive a₁ = 14, aₙ₊₁ = aₙ - 5 and explicit aₙ = 14 - 5(n - 1) = 14 - 5n + 5 = 19 - 5n. Choice A correctly writes both formulas with a₁ = 14 as the first term and d = -5 as the common difference (subtracting 5 each time). Choice B incorrectly adds 5 instead of subtracting, C has wrong first term 9, and D treats this as geometric with ratio -5. Sequence type decision: calculate differences between consecutive terms (constant → arithmetic with that d) AND ratios of consecutive terms (constant → geometric with that r). When the common difference is negative, we subtract in the recursive form (aₙ₊₁ = aₙ - |d|) and the explicit form becomes aₙ = a₁ + (n-1)d where d is negative - don't forget the signs!
A town has a population of 50,000 and grows by 3% each year. Model the population after $n$ years with both a recursive and an explicit formula, where $a_1$ is the population after 1 year (domain: positive integers).
Answer: Recursive: $a_1=51{,}500,\ a_{n+1}=1.03a_n$ ; Explicit: $a_n=51{,}500\cdot(1.03)^{n-1}$ 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 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!

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