- A fitness program requires participants to do push-ups daily. Week 1 requires 20 push-ups per day, and each subsequent week increases by 8 push-ups per day. If a participant completes exactly 10 weeks of the program, what is the average number of push-ups per day over the entire program?
Answer: 56 push-ups per day
The daily push-ups per week form an arithmetic sequence: 20, 28, 36, 44, 52, 60, 68, 76, 84, 92. The average of an arithmetic sequence equals the average of the first and last terms: (20 + 92)/2 = 112/2 = 56 push-ups per day. Alternatively, we can verify using the sum formula: S₁₀ = 10/2[2(20) + 9(8)] = 5[40 + 72] = 5(112) = 560 total push-ups per day across 10 weeks, so the average is 560/10 = 56 push-ups per day.
- A company's quarterly profits follow an arithmetic sequence. In the first quarter, they earned $45,000. By the fourth quarter of the same year, their profit had increased to $63,000. If this trend continues, what will be their total profit for the first two years (8 quarters)?
Answer: $528,000
We have a₁ = 45,000 and a₄ = 63,000. Since a₄ = a₁ + 3d, we get 63,000 = 45,000 + 3d, so 3d = 18,000, giving d = 6,000. For 8 quarters: S₈ = 8/2[2(45,000) + (8-1)(6,000)] = 4[90,000 + 42,000] = 4(132,000) = $528,000. Choice B incorrectly uses d = 6,500. Choice C uses d = 7,000. Choice D uses only 7 quarters instead of 8.
- A theater has 24 seats in the first row, 28 seats in the second row, 32 seats in the third row, and so on. If the theater has 15 rows total and tickets cost $12 per seat, what is the total revenue when the theater is completely sold out?
Answer: $7,560
First find the total number of seats using the arithmetic sequence formula. We have a₁ = 24, d = 4, n = 15. The sum is S₁₅ = 15/2[2(24) + (15-1)(4)] = 15/2[48 + 56] = 15/2(104) = 15(52) = 780 seats. Total revenue = 780 × $12 = $7,560. Choice A incorrectly uses 14 rows instead of 15. Choice B uses the wrong common difference of 6 instead of 4. Choice D results from arithmetic errors in the summation formula.
- A pyramid is built using identical cubic blocks. The top layer has 1 block, the second layer has 4 blocks, the third layer has 7 blocks, and each subsequent layer has 3 more blocks than the layer above it. How many blocks are in the bottom layer if the pyramid has 8 layers total?
Answer: 22 blocks
The blocks per layer form an arithmetic sequence: 1, 4, 7, 10, 13, 16, 19, 22 for layers 1 through 8 (top to bottom). With first term a₁ = 1 and common difference d = 3, the 8th term is a₈ = 1 + (8-1)(3) = 1 + 21 = 22 blocks. The other choices represent common errors: 19 is the 7th layer, 25 would be if d = 4 instead of 3, and 28 would be if we miscounted the layers.
- A savings plan deposits $150 in the first month, $175 in the second month, $200 in the third month, and continues this pattern. After how many months will the account first contain more than $4,000 total?
Answer: 14 months
This is an arithmetic sequence with a₁ = 150, d = 25. We need the smallest n where Sn > 4000. Using Sn = n/2[2(150) + (n-1)(25)] = n/2[300 + 25n - 25] = n/2[275 + 25n]. Setting this > 4000: n(275 + 25n)/2 > 4000, so n(275 + 25n) > 8000, giving 275n + 25n² > 8000, or 25n² + 275n - 8000 > 0, or n² + 11n - 320 > 0. Using the quadratic formula: n = (-11 + √(121 + 1280))/2 = (-11 + √1401)/2 ≈ (-11 + 37.43)/2 ≈ 13.22. Since we need the first month where this exceeds $4000, we need n = 14. Choice A results from rounding errors. Choice B uses ≥ instead of >. Choice D adds an extra month unnecessarily. - A company implements a new employee bonus structure where each employee receives an initial bonus of $200 in January, and then receives an additional $75 each subsequent month. If an employee stays with the company for exactly 18 months under this plan, what is the total amount of bonuses they will receive?
Answer: $2,825
This is an arithmetic sequence with first term a₁ = 200, common difference d = 75, and n = 18 terms. The sum formula is Sn = n/2[2a₁ + (n-1)d]. So S₁₈ = 18/2[2(200) + (18-1)(75)] = 9[400 + 17(75)] = 9[400 + 1275] = 9(1675) = $2,825. Choice A uses only 17 terms instead of 18. Choice B incorrectly adds 75 to choice A. Choice D incorrectly uses n = 18 instead of n-1 = 17 in the formula. - A medication dosage schedule starts with 400mg on day 1, then decreases by 25mg each day. On which day will the patient take their last dose before the medication is completely discontinued?
Answer: Day 16
This is an arithmetic sequence with a₁ = 400 and d = -25. The general term is aₙ = 400 + (n-1)(-25) = 400 - 25(n-1) = 425 - 25n. The medication is discontinued when the dose would be 0 or negative. Setting 425 - 25n ≤ 0: 425 ≤ 25n, so n ≥ 17. On day 16: a₁₆ = 425 - 25(16) = 425 - 400 = 25mg. On day 17: a₁₇ = 425 - 25(17) = 425 - 425 = 0mg. So day 16 is the last day with a positive dose.
