Interpreting Sequence Parameters - Math 1

1. A water tank contains 450 gallons initially. A pump removes 12 gallons every hour, while evaporation causes an additional loss of 3 gallons per hour. If the sequence $a_n = 450 - 15n$ models the water remaining after $n$ hours, which statement best explains why the coefficient 15 appears in this formula?
   Answer: It represents the combined effect of pump removal and evaporation per hour When you encounter a sequence modeling a real-world situation with multiple rates of change, you need to identify how these rates combine to affect the overall pattern. The sequence $a_n = 450 - 15n$ shows water decreasing by 15 gallons each hour. To understand where this coefficient comes from, examine what's happening: the pump removes 12 gallons per hour, and evaporation removes an additional 3 gallons per hour. Since both processes reduce the water amount simultaneously, you add their effects: $12 + 3 = 15$ gallons lost per hour total. Answer D correctly identifies that 15 represents the combined effect of both the pump removal and evaporation working together each hour. Answer A incorrectly suggests subtracting the rates ($12 - 3 = 9$), which would imply evaporation somehow counteracts the pump rather than contributing to water loss. Answer B introduces a nonexistent "25% increase due to system inefficiency" that isn't mentioned in the problem and would give $12 \times 1.25 = 15$, ignoring evaporation entirely. Answer C claims the coefficient represents "three times the evaporation rate" ($3 \times 3 = 9$), which both gives the wrong value and ignores the pump's contribution. When modeling situations with multiple rates working in the same direction, remember to add them together. The coefficient in a linear sequence always represents the total rate of change per time unit, regardless of how many individual processes contribute to that change.
2. A sequence $s_n = 45 - 5n$ models the number of seats remaining in a theater after $n$ groups of tickets are sold. A student claims that since $s_1 = 40$, the theater initially had 40 seats. What is the error in this reasoning?
   Answer: The student used $n = 1$ instead of $n = 0$ to find the initial number of seats When working with sequence formulas that model real-world situations, you need to identify what value of the variable represents the starting condition. Here, the sequence $s_n = 45 - 5n$ tells us how many seats remain after $n$ groups have purchased tickets. To find the initial number of seats (before any groups bought tickets), you should substitute $n = 0$: $s_0 = 45 - 5(0) = 45$. This means the theater originally had 45 seats. The student's error was using $n = 1$ instead of $n = 0$ to represent the starting point. When $n = 1$, one group has already purchased tickets, so $s_1 = 40$ represents the remaining seats after the first sale, not the original capacity. Choice A is incorrect because the student did understand that $s_1$ represents remaining seats, but incorrectly thought this was the initial state. Choice B misses the point—the student wasn't confused about the 5 seats per group, but about which value of $n$ to use. Choice C suggests an unnecessary calculation; you don't need to add anything to $s_0$ since $s_0$ already gives you the initial capacity directly. Remember: in sequence problems involving time or steps, $n = 0$ typically represents the starting condition, while $n = 1$ represents the state after the first change has occurred. Always check what the variable represents before determining initial conditions.
3. An arithmetic sequence has $a_3 = 17$ and $a_8 = 37$. If this sequence models the total cost (in dollars) of a service after $n$ months, including an initial setup fee, what does the difference $a_1 - d$ represent in this billing context?
   Answer: The initial setup fee charged before any monthly service begins First, find the common difference: $d = \frac{a_8 - a_3}{8-3} = \frac{37-17}{5} = 4$. Since $a_3 = 17$, we have $a_1 = 17 - 2(4) = 9$. In this context, $a_n$ represents total cumulative cost after $n$ months, so $a_1 = 9$ is the total cost after month 1 (setup fee + first month's charge). The common difference $d = 4$ is the monthly service rate. Therefore, $a_1 - d = 9 - 4 = 5$ represents just the setup fee before monthly service begins. Choice A is incorrect because that's $d$, not $a_1 - d$. Choice C is incorrect because that's $a_1$, not $a_1 - d$. Choice D is incorrect because the cost difference between consecutive months is $d$, not $a_1 - d$.
4. A sequence $S_n = 60 + 8(n-3)$ models the score of a student on practice test $n$. The student's tutor claims that since $S_3 = 60$, the student's initial ability level corresponds to a score of 60. A more careful analysis reveals that $S_1 = 44$. What does this suggest about interpreting the sequence parameters in this learning context?
   Answer: The parameter 60 represents target performance when $n = 3$, not initial ability; actual starting performance is 44 The sequence $S_n = 60 + 8(n-3)$ gives $S_3 = 60 + 8(0) = 60$ and $S_1 = 60 + 8(-2) = 44$. The tutor incorrectly interpreted the parameter 60 as representing initial ability, when it actually represents the score achieved on test 3. The true starting performance is $S_1 = 44$. The parameter 60 is the reference point when $(n-3) = 0$, but this doesn't represent initial ability—it's the performance level reached by test 3. Choice A incorrectly supports the tutor's interpretation. Choice C introduces external factors not indicated by the sequence structure. Choice D correctly identifies the improvement pattern but doesn't address the parameter interpretation error as directly as choice B.
5. An elevator starts at the 8th floor. It moves according to the pattern: up 3 floors, then down 1 floor, then up 3 floors, then down 1 floor, and so on. After this up-down pattern repeats $n$ complete times, the elevator's floor position can be modeled as $F_n = 8 + 2n$. What does the coefficient 2 represent in this context?
   Answer: The net change in floor position per complete up-down cycle Each complete cycle consists of moving up 3 floors then down 1 floor, for a net gain of $3 - 1 = 2$ floors per cycle. The coefficient 2 represents this net change per cycle. Choice B incorrectly calculates the average as $(3 + (-1))/2 = 1$. Choice C represents total distance traveled ($3 + 1 = 4$), not net change. Choice D describes what the coefficient represents mathematically but not what it means in the elevator context.
6. A sequence models daily medication dosage: $D_n = D_1 + (n-1) \cdot r$, where $D_n$ is the dosage on day $n$, $D_1 = 20$ mg, and $r = -2.5$ mg. A pharmacist calculates that the medication will be discontinued when the dosage reaches 5 mg. On which day will this occur, and what does the parameter $r$ indicate about the treatment plan?
   Answer: Day 7; the dosage decreases by 2.5 mg daily as part of a gradual withdrawal protocol This problem tests your understanding of arithmetic sequences, where each term follows the pattern $D_n = D_1 + (n-1) \cdot r$. The key is recognizing what each parameter means and solving for when the sequence reaches a target value. To find when the dosage reaches 5 mg, substitute the given values into the formula: $5 = 20 + (n-1) \cdot (-2.5)$. Solving: $5 - 20 = (n-1) \cdot (-2.5)$, so $-15 = (n-1) \cdot (-2.5)$. Dividing both sides by -2.5 gives $n-1 = 6$, therefore $n = 7$. The parameter $r = -2.5$ represents the common difference—the amount the dosage decreases each day. In medical contexts, a systematic daily reduction often indicates a tapering protocol to safely withdraw patients from medication. Answer A incorrectly calculates day 9 and misinterprets the parameter as relating to medication effectiveness rather than dosage changes. Answer B correctly identifies the daily 2.5 mg decrease but miscalculates the day as 6 instead of 7, and assumes the reduction is for minimizing side effects during active treatment. Answer C also miscalculates the day as 8 and suggests reaching a "minimum effective dose" rather than discontinuation. Answer D correctly identifies day 7 and properly interprets the negative parameter as part of a gradual withdrawal protocol, which is the most appropriate medical context for systematically reducing dosage to zero. When working with arithmetic sequences in applied contexts, always pay attention to what the common difference represents in real-world terms—it often reveals the underlying process or strategy being modeled.
7. A machine produces widgets at a constant rate. At 2:00 PM, there were 340 widgets completed. At 5:00 PM, there were 445 widgets completed. If the production follows the pattern $W_t = 340 + 35t$, where $W_t$ is the number of widgets $t$ hours after 2:00 PM, what assumption about production is built into this model?
   Answer: The machine produces exactly 35 widgets per hour with no breaks or slowdowns The linear model $W_t = 340 + 35t$ assumes constant production of 35 widgets per hour. We can verify: at $t = 0$ (2:00 PM), $W_0 = 340$. At $t = 3$ (5:00 PM), $W_3 = 340 + 35(3) = 445$. Choice B misinterprets the starting value - it represents total widgets at 2:00 PM, not when production began. Choice C incorrectly suggests the rate increases over time, but 35 is the constant rate, not an increasing rate. Choice D suggests batch production rather than continuous production at a constant rate.
8. A sequence $a_n = 12 + 4(n-2)$ represents the number of hours of daylight on the $n$th day after the winter solstice. A student states that this sequence shows daylight increases by 4 hours per day starting from 8 hours on day 1. Which part of this interpretation needs correction?
   Answer: The starting amount should be 12 hours, not 8 hours, based on the sequence parameters From $a_n = 12 + 4(n-2)$, the first day gives $a_1 = 12 + 4(1-2) = 12 + 4(-1) = 12 - 4 = 8$ hours. The student correctly identified 8 hours on day 1 and the 4-unit daily increase. However, the student incorrectly stated the starting amount as 8 hours when the sequence parameter 12 represents the base value before the $(n-2)$ adjustment. At $n = 2$: $a_2 = 12 + 4(0) = 12$ hours. So day 2 has 12 hours, which is the base parameter value. Choice A introduces external realism not indicated in the mathematical problem. Choice C is incorrect because $n = 1$ gives a valid result. Choice D misinterprets the $(n-2)$ structure, which still represents a daily change with a shifted reference point.
9. A sequence is defined by $c_n = -3n + k$ where $k$ is a constant. If $c_5 = 12$ and this sequence models the temperature (°F) inside a freezer during hour $n$ of a cooling process, what does the parameter $k$ represent in this context?
   Answer: The initial temperature when the cooling process began at hour 0 First, find $k$: Since $c_5 = 12$, we have $-3(5) + k = 12$, so $-15 + k = 12$, thus $k = 27$. The sequence $c_n = -3n + 27$ can be interpreted as starting from temperature 27°F at time $n = 0$ (before the process begins), then decreasing by 3°F each hour. So $k = 27$ represents the initial temperature when $n = 0$. Choice B is incorrect because $c_1 = -3(1) + 27 = 24$, not $k$. Choice C is incorrect because the rate of decrease is 3°F/hour (the coefficient of $n$), not $k$. Choice D is incorrect because this sequence continues decreasing indefinitely, with no final temperature.
10. A bacteria colony starts with 80 bacteria. Every 3 hours, the population increases by 25 bacteria. After how many complete 3-hour periods will the population first reach at least 230 bacteria?
    Answer: 6 periods This is a linear growth problem where you need to find when a population reaches a target value. When you see questions about populations growing by a fixed amount at regular intervals, set up an equation to track the pattern over time. The bacteria colony starts with 80 bacteria and gains 25 bacteria every 3-hour period. After $n$ complete periods, the population will be: $80 + 25n$. You need to find the smallest whole number $n$ where this population first reaches at least 230 bacteria. Set up the inequality: $80 + 25n \geq 230$. Solving: $25n \geq 150$, so $n \geq 6$. Let's verify: after 6 periods, the population is $80 + 25(6) = 80 + 150 = 230$ bacteria. This exactly meets the requirement of "at least 230," so the answer is 6 periods. Choice A (5 periods) gives you $80 + 25(5) = 205$ bacteria, which falls short of the 230 target. Choice C (7 periods) gives you $80 + 25(7) = 255$ bacteria, which exceeds the requirement but isn't the first time you reach at least 230. Choice B (8 periods) gives you $80 + 25(8) = 280$ bacteria, which is even further beyond what's needed and definitely not the first occurrence. The correct answer is D) 6 periods. For linear growth problems, always set up the equation $\text{initial value} + \text{rate} \times \text{time periods}$, then solve the inequality for the minimum number of periods needed. Don't forget to check that your answer actually satisfies the "at least" condition.
11. A savings account has an initial balance and earns a fixed amount each month through interest and automatic deposits. The balance after $n$ months is $B_n = 1200 + 75n$ dollars. If someone wants to reach exactly $\$2175$ in their account, but can only make withdrawals at the end of complete months, what is the latest complete month they should make their withdrawal?
    Answer: Month 13 Setting $B_n = 2175$: $1200 + 75n = 2175$, so $75n = 975$, giving $n = 13$. At the end of month 13, the balance will be exactly $2175. At the end of month 12: $B_{12} = 1200 + 75(12) = 2100$, which is less than the target. At the end of month 14: $B_{14} = 1200 + 75(14) = 2250$, which exceeds the target. Since the question asks for the latest complete month to reach exactly $2175, the answer is month 13.
12. A subscription service charges an initial setup fee plus a monthly rate. The total amount paid after $n$ months is given by $C_n = 45 + 18n$ dollars. If a customer has paid exactly $\$171$ total, for how many complete months have they been subscribed?
    Answer: 7 months This problem involves a linear cost function where you need to work backwards from a total cost to find the number of months. The equation $C_n = 45 + 18n$ shows that there's a $45 setup fee plus $18 per month. To find how many months correspond to a total payment of $171, you set up the equation: $171 = 45 + 18n$. Subtracting the setup fee from both sides gives you $126 = 18n$. Dividing both sides by 18 yields $n = 7$. Let's verify: After 7 months, the total cost would be $C_7 = 45 + 18(7) = 45 + 126 = 171$. This matches perfectly. Looking at the wrong answers: Choice (A) 6 months would give $C_6 = 45 + 18(6) = 45 + 108 = 153$, which is $18 short of the target. Choice (B) 8 months would yield $C_8 = 45 + 18(8) = 45 + 144 = 189$, which exceeds the target by $18. Choice (D) 9 months would give $C_9 = 45 + 18(9) = 45 + 162 = 207$, which is $36 over the target. The correct answer is (C) 7 months. Study tip: When solving linear equations in real-world contexts, always verify your answer by substituting back into the original equation. This catches arithmetic errors and confirms your solution makes sense within the problem's context.
13. A parking meter starts with 120 minutes of time. Every 15 minutes, it loses 15 minutes from the display, but a maintenance fee also reduces the remaining time by an additional 2 minutes. If $T_n$ represents the time showing after $n$ fifteen-minute intervals, what does the parameter $-17$ represent in the formula $T_n = 120 - 17n$?
    Answer: The total time lost per interval including the maintenance fee In each 15-minute interval, the meter loses 15 minutes of actual time passage plus an additional 2 minutes due to the maintenance fee, for a total reduction of 17 minutes per interval. The parameter $-17$ represents this total decrease per interval. Choice B incorrectly describes it as a difference rather than a sum. Choice C incorrectly relates it to the initial time. Choice D is wrong because it suggests maintenance costs are excluded, when they are actually included in the -17.
14. Two arithmetic sequences represent different investment strategies: Sequence A: $a_n = 500 + 75(n-1)$ and Sequence B: $b_n = 800 + 25(n-1)$, where $n$ represents months. An investor wants to know after how many complete months Strategy A will first exceed Strategy B. What equation correctly models this scenario?
    Answer: $500 + 75(n-1) > 800 + 25(n-1)$, solved for the smallest integer $n$ The problem asks when Strategy A first exceeds Strategy B after complete months. This requires comparing $a_n > b_n$, which gives $500 + 75(n-1) > 800 + 25(n-1)$. Solving: $500 + 75n - 75 > 800 + 25n - 25$, so $425 + 75n > 775 + 25n$, thus $50n > 350$, giving $n > 7$. So Strategy A first exceeds Strategy B after month 8. Choice B is incorrect because it uses $75n$ instead of $75(n-1)$, changing the sequence structure. Choice C is incorrect because it rearranges to compare only the variable parts, losing the base values. Choice D is incorrect because it uses different indexing for the two sequences.
15. The number of active users (in millions) after $n$ quarters is given by $U_n = 2.5 + 0.8n$. If the platform started tracking when it had 4.1 million users, which quarter does this correspond to in the model?
    Answer: Quarter 2 We need to find $n$ when $U_n = 4.1$. Setting up the equation: $2.5 + 0.8n = 4.1$, so $0.8n = 1.6$, which gives $n = 2$. This means 4.1 million users corresponds to Quarter 2 in the model. Choice A gives $U_1 = 2.5 + 0.8(1) = 3.3$ million. Choice C gives $U_3 = 2.5 + 0.8(3) = 4.9$ million. Choice D gives $U_4 = 2.5 + 0.8(4) = 5.7$ million.
16. A delivery truck's odometer reads 15,420 miles at the start of a route. The truck averages 55 miles per hour and completes one delivery every hour, adding 8 miles of local driving per delivery. After $h$ hours, the odometer reading is $M_h = 15420 + 63h$. What real-world factors contribute to the coefficient 63?
    Answer: Average speed on highways plus additional mileage for local delivery routes When you encounter linear functions in word problems, focus on breaking down what each part of the equation represents. The function $M_h = 15420 + 63h$ shows how the odometer reading changes over time, where the coefficient 63 tells you how many miles are added each hour. To find what contributes to this coefficient, you need to identify all sources of mileage per hour. The truck averages 55 miles per hour on its route, which accounts for the primary distance covered. Additionally, each delivery requires 8 miles of local driving (turns, parking, navigating to specific addresses), and since one delivery happens per hour, that's 8 additional miles per hour. Therefore: 55 + 8 = 63 miles per hour total. Answer C correctly identifies these two components: "Average speed on highways plus additional mileage for local delivery routes." Answer A mentions "fuel efficiency calculations," but fuel efficiency doesn't affect odometer readings—it only impacts gas consumption. Answer B refers to "overtime mileage adjustments," which isn't mentioned in the problem and doesn't relate to the regular hourly pattern described. Answer D suggests "traffic delay compensation," but delays would reduce average speed rather than add extra mileage to the odometer. Remember that in linear function word problems, coefficients represent rates of change. Always identify every factor that contributes to that rate—in this case, both highway travel and local delivery driving add to the total miles per hour.
17. An experiment measures the pH of a solution over time using the model $pH_n = 6.8 + 0.3(n-1)$, where $n$ represents the number of minutes elapsed. A lab report states that the solution becomes neutral (pH = 7) after 1 minute. Based on the sequence parameters, what is the most accurate assessment of this claim?
    Answer: Incorrect, because neutrality occurs at $n = \frac{5}{3}$ minutes, not exactly 1 minute To find when pH = 7: $7 = 6.8 + 0.3(n-1)$, so $0.2 = 0.3(n-1)$, giving $n-1 = \frac{0.2}{0.3} = \frac{2}{3}$, thus $n = 1 + \frac{2}{3} = \frac{5}{3} ≈ 1.67$ minutes. At $n = 1$: $pH_1 = 6.8 + 0.3(0) = 6.8$, which is acidic. Choice A is incorrect because 6.8 doesn't round to 7 in scientific contexts where precision matters. Choice B is partially correct but doesn't provide the complete analysis. Choice D is incorrect because 0.3 units per minute is a reasonable rate, and neutrality is reached in about 1.67 minutes.
18. A savings account balance follows the arithmetic sequence $B_n = 1200 + 150(n-1)$, where $B_n$ is the balance at the end of month $n$. A financial advisor claims that the account holder deposits $150 each month starting with an initial deposit of $1200. Which aspect of this interpretation is incorrect?
    Answer: The sequence represents total balance, not individual deposits, so the initial amount was already in the account The sequence $B_n = 1200 + 150(n-1)$ shows $B_1 = 1200$, meaning the account has $1200 at the end of month 1. The common difference $d = 150$ means the balance increases by $150 each month. However, since $B_1 = 1200$ is the balance after month 1, this suggests the account started with some amount and then $150 was added during month 1, resulting in a $1200 balance. The advisor incorrectly assumes $1200 was an initial deposit when it's actually the balance after the first month's activity. Choice A is incorrect because $150 is the correct monthly increase. Choice B is incorrect because $1200 is the correct end-of-month-1 balance. Choice D is incorrect because the problem states arithmetic sequence, implying constant additions, not compound interest.
19. A research study tracks the population of a bacterial colony that decreases by a constant amount each hour due to treatment. The population follows the pattern: $P_n = 850 - 45(n-1)$, where $P_n$ represents the population after $n$ hours of treatment. A lab technician incorrectly states that the initial population was 805 bacteria. What is the most likely source of this error?
    Answer: The technician calculated $P_1 - 45$ instead of evaluating $P_1$ directly from the formula The correct initial population is $P_1 = 850 - 45(1-1) = 850 - 45(0) = 850$. The technician's answer of 805 equals $850 - 45 = P_1 - 45$, suggesting they incorrectly subtracted the rate from the initial value. Choice B is incorrect because $P_0 = 850 - 45(-1) = 895$, not 805. Choice C is incorrect because the decrease rate is 45, not 805. Choice D is incorrect because $850 + 45 = 895$, not 805.
20. A subscription service charges an initial setup fee plus a fixed monthly rate. After 3 months, a customer has paid $127 total. After 8 months, the same customer has paid $252 total. If this payment pattern represents an arithmetic sequence where $a_n$ is the total amount paid after $n$ months, what does the parameter $a_1 - d$ represent in this context?
    Answer: The initial setup fee charged before any monthly payments First, find the common difference: $d = \frac{252-127}{8-3} = \frac{125}{5} = 25$. Since $a_3 = 127$, we have $a_1 = 127 - 2(25) = 77$. In this sequence, $a_n$ represents total cumulative payments, so $a_1 = 77$ is the total paid after month 1 (setup fee + first month's rate). Therefore, $a_1 - d = 77 - 25 = 52$ represents just the initial setup fee before any monthly payments begin. Choice A is incorrect because $d = 25$ is the monthly rate. Choice C is incorrect because this represents an actual fee, not savings. Choice D is incorrect because that's $a_1$, not $a_1 - d$.

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