Math 1 Quiz: Interpreting Sequence Parameters
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Interpreting Sequence ParametersQuestion 1 of 20

Two arithmetic sequences represent inventory levels: In=200−15(n−1)I_n = 200 - 15(n-1) and Jn=80+25(n−1)J_n = 80 + 25(n-1), where nn is the week number. A manager notes that both inventories will be equal when 200−15(n−1)=80+25(n−1)200 - 15(n-1) = 80 + 25(n-1). Solving this equation gives n=4n = 4. What do the sequence parameters tell us about the inventory trends leading to this intersection?

Both inventories change by 40 units total weekly, but in opposite directions from their starting points
Inventory I decreases by 15 units weekly from 185, while J increases by 25 units weekly from 105
Inventory I starts higher but decreases faster, while J starts lower but increases steadily
Inventory I decreases by 15 units weekly from 200, while J increases by 25 units weekly from 80
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Math 1 Quiz

Math 1 Quiz: Interpreting Sequence Parameters

Practice Interpreting Sequence Parameters in Math 1 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Interpreting Sequence Parameters, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 1.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

Two arithmetic sequences represent inventory levels: In=200−15(n−1)I_n = 200 - 15(n-1) and Jn=80+25(n−1)J_n = 80 + 25(n-1), where nn is the week number. A manager notes that both inventories will be equal when 200−15(n−1)=80+25(n−1)200 - 15(n-1) = 80 + 25(n-1). Solving this equation gives n=4n = 4. What do the sequence parameters tell us about the inventory trends leading to this intersection?

  1. Both inventories change by 40 units total weekly, but in opposite directions from their starting points
  2. Inventory I decreases by 15 units weekly from 185, while J increases by 25 units weekly from 105
  3. Inventory I starts higher but decreases faster, while J starts lower but increases steadily
  4. Inventory I decreases by 15 units weekly from 200, while J increases by 25 units weekly from 80 (correct answer)
Explanation: When you encounter arithmetic sequences in word problems, focus on identifying the first term and common difference from the given formulas. These parameters directly reveal the behavior patterns you need to analyze. For arithmetic sequences in the form an=a1+d(n−1)a_n = a_1 + d(n-1), the first term a1a_1 is the starting value, and dd is the common difference (how much the sequence changes each step). Looking at In=200−15(n−1)I_n = 200 - 15(n-1), you can rewrite this as In=200+(−15)(n−1)I_n = 200 + (-15)(n-1). This shows inventory I starts at 200 units when n=1n=1 and decreases by 15 units each week (negative common difference). For Jn=80+25(n−1)J_n = 80 + 25(n-1), inventory J starts at 80 units and increases by 25 units weekly (positive common difference). Answer D correctly identifies these exact parameters: I decreases by 15 units weekly from its starting point of 200, while J increases by 25 units weekly from its starting point of 80. Answer A incorrectly states "40 units total" and mentions "opposite directions from starting points," which misinterprets how to read the parameters. Answer B gives wrong starting values (185 and 105 instead of 200 and 80) by apparently substituting n=2n=2. Answer C provides general trends but lacks the specific numerical values that make this analysis meaningful. Study tip: Always identify the first term by substituting n=1n=1 into arithmetic sequence formulas, and read the coefficient of (n−1)(n-1) as your common difference. The sign tells you the direction of change.

Question 2

A savings account balance follows the arithmetic sequence Bn=1200+150(n−1)B_n = 1200 + 150(n-1), where BnB_n is the balance at the end of month nn. 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?

  1. The monthly deposit amount should be $1050, not $150, based on the sequence parameters
  2. The initial deposit amount should be $1050, not $1200, based on the sequence structure
  3. The sequence represents total balance, not individual deposits, so the initial amount was already in the account (correct answer)
  4. The sequence shows $150 is added each month, but this includes interest, not just deposits
Explanation: The sequence Bn=1200+150(n−1)B_n = 1200 + 150(n-1) shows B1=1200B_1 = 1200, meaning the account has $1200 at the end of month 1. The common difference $d=150d = 150 meansthebalanceincreasesbymeans the balance increases by150 each month. However, since B1=1200B_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.

Question 3

A sequence sn=45−5ns_n = 45 - 5n models the number of seats remaining in a theater after nn groups of tickets are sold. A student claims that since s1=40s_1 = 40, the theater initially had 40 seats. What is the error in this reasoning?

  1. The student confused the number of remaining seats with the original seating capacity
  2. The student forgot to account for the 5 seats sold to each group of customers
  3. The student should have calculated s0+5s_0 + 5 to find the total seating capacity
  4. The student used n=1n = 1 instead of n=0n = 0 to find the initial number of seats (correct answer)
Explanation: 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 sn=45−5ns_n = 45 - 5n tells us how many seats remain after nn groups have purchased tickets. To find the initial number of seats (before any groups bought tickets), you should substitute n=0n = 0: s0=45−5(0)=45s_0 = 45 - 5(0) = 45. This means the theater originally had 45 seats. The student's error was using n=1n = 1 instead of n=0n = 0 to represent the starting point. When n=1n = 1, one group has already purchased tickets, so s1=40s_1 = 40 represents the remaining seats after the first sale, not the original capacity. Choice A is incorrect because the student did understand that s1s_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 nn to use. Choice C suggests an unnecessary calculation; you don't need to add anything to s0s_0 since s0s_0 already gives you the initial capacity directly. Remember: in sequence problems involving time or steps, n=0n = 0 typically represents the starting condition, while n=1n = 1 represents the state after the first change has occurred. Always check what the variable represents before determining initial conditions.

Question 4

Two arithmetic sequences represent different investment strategies: Sequence A: an=500+75(n−1)a_n = 500 + 75(n-1) and Sequence B: bn=800+25(n−1)b_n = 800 + 25(n-1), where nn 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?

  1. 500+75(n−1)>800+25(n−1)500 + 75(n-1) > 800 + 25(n-1), solved for the smallest integer nn (correct answer)
  2. 500+75n>800+25n500 + 75n > 800 + 25n, solved for the smallest integer nn
  3. 75(n−1)−25(n−1)>800−50075(n-1) - 25(n-1) > 800 - 500, solved for the smallest integer nn
  4. 500+75n>800+25(n+1)500 + 75n > 800 + 25(n+1), solved for the smallest integer nn
Explanation: The problem asks when Strategy A first exceeds Strategy B after complete months. This requires comparing an>bna_n > b_n, which gives 500+75(n−1)>800+25(n−1)500 + 75(n-1) > 800 + 25(n-1). Solving: 500+75n−75>800+25n−25500 + 75n - 75 > 800 + 25n - 25, so 425+75n>775+25n425 + 75n > 775 + 25n, thus 50n>35050n > 350, giving n>7n > 7. So Strategy A first exceeds Strategy B after month 8. Choice B is incorrect because it uses 75n75n instead of 75(n−1)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.

Question 5

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: Pn=850−45(n−1)P_n = 850 - 45(n-1), where PnP_n represents the population after nn hours of treatment. A lab technician incorrectly states that the initial population was 805 bacteria. What is the most likely source of this error?

  1. The technician calculated P1−45P_1 - 45 instead of evaluating P1P_1 directly from the formula (correct answer)
  2. The technician used n=0n = 0 instead of n=1n = 1 when finding the starting population
  3. The technician confused the constant decrease rate with the initial population value
  4. The technician added the decrease rate to 850 instead of subtracting it appropriately
Explanation: The correct initial population is P1=850−45(1−1)=850−45(0)=850P_1 = 850 - 45(1-1) = 850 - 45(0) = 850. The technician's answer of 805 equals 850−45=P1−45850 - 45 = P_1 - 45, suggesting they incorrectly subtracted the rate from the initial value. Choice B is incorrect because P0=850−45(−1)=895P_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=895850 + 45 = 895, not 805.

Question 6

A sequence an=12+4(n−2)a_n = 12 + 4(n-2) represents the number of hours of daylight on the nnth 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?

  1. The daily increase should be 4 minutes, not 4 hours, based on realistic daylight changes
  2. The starting amount should be 12 hours, not 8 hours, based on the sequence parameters (correct answer)
  3. The sequence index should begin at n=0n = 0, not n=1n = 1, for proper interpretation
  4. The increase occurs every 2 days, not daily, based on the (n−2)(n-2) term structure
Explanation: From an=12+4(n−2)a_n = 12 + 4(n-2), the first day gives a1=12+4(1−2)=12+4(−1)=12−4=8a_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)(n-2) adjustment. At n=2n = 2: a2=12+4(0)=12a_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=1n = 1 gives a valid result. Choice D misinterprets the (n−2)(n-2) structure, which still represents a daily change with a shifted reference point.

Question 7

A sequence models daily medication dosage: Dn=D1+(n−1)⋅rD_n = D_1 + (n-1) \cdot r, where DnD_n is the dosage on day nn, D1=20D_1 = 20 mg, and r=−2.5r = -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 rr indicate about the treatment plan?

  1. Day 9; the negative parameter indicates the medication becomes less effective over time
  2. Day 6; the dosage decreases by 2.5 mg daily to minimize side effects during treatment
  3. Day 8; the dosage is reduced by 2.5 mg daily to reach the minimum effective dose
  4. Day 7; the dosage decreases by 2.5 mg daily as part of a gradual withdrawal protocol (correct answer)
Explanation: This problem tests your understanding of arithmetic sequences, where each term follows the pattern Dn=D1+(n−1)⋅rD_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)⋅(−2.5)5 = 20 + (n-1) \cdot (-2.5). Solving: 5−20=(n−1)⋅(−2.5)5 - 20 = (n-1) \cdot (-2.5), so −15=(n−1)⋅(−2.5)-15 = (n-1) \cdot (-2.5). Dividing both sides by -2.5 gives n−1=6n-1 = 6, therefore n=7n = 7. The parameter r=−2.5r = -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.

Question 8

A sequence Sn=60+8(n−3)S_n = 60 + 8(n-3) models the score of a student on practice test nn. The student's tutor claims that since S3=60S_3 = 60, the student's initial ability level corresponds to a score of 60. A more careful analysis reveals that S1=44S_1 = 44. What does this suggest about interpreting the sequence parameters in this learning context?

  1. The tutor correctly identified the baseline ability; the score of 44 represents performance before adequate preparation
  2. The parameter 60 represents target performance when n=3n = 3, not initial ability; actual starting performance is 44 (correct answer)
  3. Both scores are relevant: 60 is the natural ability level, while 44 includes test anxiety effects
  4. The sequence structure shows 60 as the expected score after 3 practice sessions, with 8-point improvements per session
Explanation: The sequence Sn=60+8(n−3)S_n = 60 + 8(n-3) gives S3=60+8(0)=60S_3 = 60 + 8(0) = 60 and S1=60+8(−2)=44S_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 S1=44S_1 = 44. The parameter 60 is the reference point when (n−3)=0(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.

Question 9

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 $ana_n isthetotalamountpaidafteris the total amount paid after nn months,whatdoestheparametermonths, what does the parameter a1−da_1 - d $ represent in this context?

  1. The monthly rate charged by the subscription service
  2. The initial setup fee charged before any monthly payments (correct answer)
  3. The total amount saved compared to paying monthly without setup
  4. The amount paid in the first month including the setup fee
Explanation: First, find the common difference: d=252−1278−3=1255=25d = \frac{252-127}{8-3} = \frac{125}{5} = 25. Since a3=127a_3 = 127, we have a1=127−2(25)=77a_1 = 127 - 2(25) = 77. In this sequence, ana_n represents total cumulative payments, so a1=77a_1 = 77 is the total paid after month 1 (setup fee + first month's rate). Therefore, a1−d=77−25=52a_1 - d = 77 - 25 = 52 represents just the initial setup fee before any monthly payments begin. Choice A is incorrect because d=25d = 25 is the monthly rate. Choice C is incorrect because this represents an actual fee, not savings. Choice D is incorrect because that's a1a_1, not a1−da_1 - d.

Question 10

An arithmetic sequence has a3=17a_3 = 17 and a8=37a_8 = 37. If this sequence models the total cost (in dollars) of a service after nn months, including an initial setup fee, what does the difference a1−da_1 - d represent in this billing context?

  1. The monthly service rate charged after the initial setup period
  2. The initial setup fee charged before any monthly service begins (correct answer)
  3. The total amount paid during the first month including all fees
  4. The cost difference between the first and second months of service
Explanation: First, find the common difference: d=a8−a38−3=37−175=4d = \frac{a_8 - a_3}{8-3} = \frac{37-17}{5} = 4. Since a3=17a_3 = 17, we have a1=17−2(4)=9a_1 = 17 - 2(4) = 9. In this context, ana_n represents total cumulative cost after nn months, so a1=9a_1 = 9 is the total cost after month 1 (setup fee + first month's charge). The common difference d=4d = 4 is the monthly service rate. Therefore, a1−d=9−4=5a_1 - d = 9 - 4 = 5 represents just the setup fee before monthly service begins. Choice A is incorrect because that's dd, not a1−da_1 - d. Choice C is incorrect because that's a1a_1, not a1−da_1 - d. Choice D is incorrect because the cost difference between consecutive months is dd, not a1−da_1 - d.

Question 11

A sequence is defined by cn=−3n+kc_n = -3n + k where kk is a constant. If c5=12c_5 = 12 and this sequence models the temperature (°F) inside a freezer during hour nn of a cooling process, what does the parameter kk represent in this context?

  1. The initial temperature when the cooling process began at hour 0 (correct answer)
  2. The temperature at the end of the first hour of cooling
  3. The rate at which temperature decreases each hour during cooling
  4. The final temperature the freezer will reach after extended cooling
Explanation: First, find kk: Since c5=12c_5 = 12, we have −3(5)+k=12-3(5) + k = 12, so −15+k=12-15 + k = 12, thus k=27k = 27. The sequence cn=−3n+27c_n = -3n + 27 can be interpreted as starting from temperature 27°F at time n=0n = 0 (before the process begins), then decreasing by 3°F each hour. So k=27k = 27 represents the initial temperature when n=0n = 0. Choice B is incorrect because c1=−3(1)+27=24c_1 = -3(1) + 27 = 24, not kk. Choice C is incorrect because the rate of decrease is 3°F/hour (the coefficient of nn), not kk. Choice D is incorrect because this sequence continues decreasing indefinitely, with no final temperature.

Question 12

An experiment measures the pH of a solution over time using the model pHn=6.8+0.3(n−1)pH_n = 6.8 + 0.3(n-1), where nn 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?

  1. Correct, because substituting n=1n = 1 gives pH1=6.8pH_1 = 6.8, which rounds to 7
  2. Incorrect, because pH1=6.8pH_1 = 6.8, which is still acidic, not neutral
  3. Incorrect, because neutrality occurs at n=53n = \frac{5}{3} minutes, not exactly 1 minute (correct answer)
  4. Incorrect, because the pH increases too slowly to reach 7 within the first minute
Explanation: To find when pH = 7: 7=6.8+0.3(n−1)7 = 6.8 + 0.3(n-1), so 0.2=0.3(n−1)0.2 = 0.3(n-1), giving n−1=0.20.3=23n-1 = \frac{0.2}{0.3} = \frac{2}{3}, thus n=1+23=53≈1.67n = 1 + \frac{2}{3} = \frac{5}{3} ≈ 1.67 minutes. At n=1n = 1: pH1=6.8+0.3(0)=6.8pH_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.

Question 13

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?

  1. 5 periods
  2. 8 periods
  3. 7 periods
  4. 6 periods (correct answer)
Explanation: 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 nn complete periods, the population will be: 80+25n80 + 25n. You need to find the smallest whole number nn where this population first reaches at least 230 bacteria. Set up the inequality: 80+25n≥23080 + 25n \geq 230. Solving: 25n≥15025n \geq 150, so n≥6n \geq 6. Let's verify: after 6 periods, the population is 80+25(6)=80+150=23080 + 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)=20580 + 25(5) = 205 bacteria, which falls short of the 230 target. Choice C (7 periods) gives you 80+25(7)=25580 + 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)=28080 + 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 initial value+rate×time periods\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.

Question 14

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 an=450−15na_n = 450 - 15n models the water remaining after nn hours, which statement best explains why the coefficient 15 appears in this formula?

  1. It represents the pump rate minus the evaporation rate per hour
  2. It represents the pump rate plus a 25% increase due to system inefficiency
  3. It represents three times the evaporation rate to account for temperature variations
  4. It represents the combined effect of pump removal and evaporation per hour (correct answer)
Explanation: 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 an=450−15na_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=1512 + 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=912 - 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 12×1.25=1512 \times 1.25 = 15, ignoring evaporation entirely. Answer C claims the coefficient represents "three times the evaporation rate" (3×3=93 \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.

Question 15

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 hh hours, the odometer reading is Mh=15420+63hM_h = 15420 + 63h. What real-world factors contribute to the coefficient 63?

  1. Highway speed plus delivery mileage plus fuel efficiency calculations
  2. Base delivery distance plus overtime mileage adjustments per hour
  3. Average speed on highways plus additional mileage for local delivery routes (correct answer)
  4. Standard route mileage plus traffic delay compensation per hour
Explanation: When you encounter linear functions in word problems, focus on breaking down what each part of the equation represents. The function Mh=15420+63hM_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.

Question 16

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 Wt=340+35tW_t = 340 + 35t, where WtW_t is the number of widgets tt hours after 2:00 PM, what assumption about production is built into this model?

  1. The machine produces exactly 35 widgets per hour with no breaks or slowdowns (correct answer)
  2. The machine started production at 2:00 PM with an initial batch of 340 widgets
  3. The machine's production rate increases by 35 widgets each hour due to efficiency gains
  4. The machine produces widgets in batches of 35 every hour starting from 340 total
Explanation: The linear model Wt=340+35tW_t = 340 + 35t assumes constant production of 35 widgets per hour. We can verify: at t=0t = 0 (2:00 PM), W0=340W_0 = 340. At t=3t = 3 (5:00 PM), W3=340+35(3)=445W_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.

Question 17

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 TnT_n represents the time showing after nn fifteen-minute intervals, what does the parameter −17-17 represent in the formula Tn=120−17nT_n = 120 - 17n?

  1. The total time lost per interval including the maintenance fee (correct answer)
  2. The maintenance fee charged every interval minus the base time reduction
  3. The initial time on the meter plus the base time reduction per interval
  4. The net change in displayed time per interval excluding maintenance costs
Explanation: 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 −17-17 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.

Question 18

A savings account has an initial balance and earns a fixed amount each month through interest and automatic deposits. The balance after nn months is Bn=1200+75nB_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?

  1. Month 12
  2. Month 13 (correct answer)
  3. Month 14
  4. Month 15
Explanation: Setting Bn=2175B_n = 2175: 1200+75n=21751200 + 75n = 2175, so 75n=97575n = 975, giving n=13n = 13. At the end of month 13, the balance will be exactly $2175. At the end of month 12: $B12=1200+75(12)=2100B_{12} = 1200 + 75(12) = 2100 ,whichislessthanthetarget.Attheendofmonth14:, which is less than the target. At the end of month 14: B14=1200+75(14)=2250B_{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.

Question 19

A research study tracks the number of active users on a social media platform over several quarters. The data follows a specific pattern that can be modeled mathematically.

The number of active users (in millions) after nn quarters is given by Un=2.5+0.8nU_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?

  1. Quarter 1
  2. Quarter 2 (correct answer)
  3. Quarter 3
  4. Quarter 4
Explanation: We need to find nn when Un=4.1U_n = 4.1. Setting up the equation: 2.5+0.8n=4.12.5 + 0.8n = 4.1, so 0.8n=1.60.8n = 1.6, which gives n=2n = 2. This means 4.1 million users corresponds to Quarter 2 in the model. Choice A gives U1=2.5+0.8(1)=3.3U_1 = 2.5 + 0.8(1) = 3.3 million. Choice C gives U3=2.5+0.8(3)=4.9U_3 = 2.5 + 0.8(3) = 4.9 million. Choice D gives U4=2.5+0.8(4)=5.7U_4 = 2.5 + 0.8(4) = 5.7 million.

Question 20

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 nn complete times, the elevator's floor position can be modeled as Fn=8+2nF_n = 8 + 2n. What does the coefficient 2 represent in this context?

  1. The net change in floor position per complete up-down cycle (correct answer)
  2. The average of the up movement and down movement per cycle
  3. The total number of floors traveled per complete up-down cycle
  4. The difference between consecutive floor positions in the sequence
Explanation: Each complete cycle consists of moving up 3 floors then down 1 floor, for a net gain of 3−1=23 - 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(3 + (-1))/2 = 1. Choice C represents total distance traveled (3+1=43 + 1 = 4), not net change. Choice D describes what the coefficient represents mathematically but not what it means in the elevator context.