- Sarah makes this pattern: 1, 4, 9, 16, 25. Jake makes this pattern: 2, 6, 12, 20, 30. What do both patterns have in common for their next number?
Answer: Both next numbers will be exactly 6 apart from each other
Sarah's pattern: 1, 4, 9, 16, 25 are perfect squares (1², 2², 3², 4², 5²), so the next is 6²=36. Jake's pattern: 2=1×2, 6=2×3, 12=3×4, 20=4×5, 30=5×6, so the next is 6×7=42. The difference between 42 and 36 is exactly 6. Choice A is wrong because 36 is even but 42-36≠6 (wait, it is 6, but 36 isn't odd). Choice B is wrong because 36 ends in 6 and 42 ends in 2. Choice D is wrong because 36 is even, not odd.
- Tom notices that 6×2=12, 6×3=18, 6×4=24. He says the next three answers will be 30, 36, 42. Is Tom correct, and what pattern do these answers follow?
Answer: Yes, Tom is correct, and the answers increase by 6 each time
Tom's calculations: 6×5=30, 6×6=36, 6×7=42 are all correct. Looking at the pattern in the products: 12, 18, 24, 30, 36, 42 - each increases by 6. This makes sense because we're adding 6 each time we multiply by the next integer (since 6×(n+1) = 6×n + 6). Choice B is wrong because Tom didn't make an error. Choice C is wrong because the increase is 6, not 12. Choice D is wrong because there's no error and the increase isn't 8.
- Based on the bar graph shown, how many oranges should Sunday have to keep the pattern?
Answer: 6
Bars rise by 1 orange each day: Wednesday 2, Thursday 3, Friday 4, Saturday 5, so Sunday must have 6. Options A, B, and C give too few oranges to continue the +1 pattern.
- In a skip counting pattern, the numbers are 8, 14, 20, 26, 32. If this pattern continues backwards before 8, what would be the three numbers that come before 8?
Answer: The three numbers before 8 are 2, -4, and -10
The pattern increases by 6 each time: 8→14(+6)→20(+6)→26(+6)→32(+6). Going backwards, we subtract 6 each time: 8→2(-6)→-4(-6)→-10(-6). So the three numbers before 8 are 2, -4, and -10. Choice B suggests the pattern decreases by 2, which doesn't match. Choice C has incorrect arithmetic. Choice D suggests decreasing by 4, which is also incorrect.
- Look at this pattern: 3, 7, 15, 31, 63. Emma says each number is double the previous number plus 1. Luis says each number is 2ⁿ⁺¹ - 1 where n is the position. Who is correct and what is the 6th number?
Answer: Both are correct, and the 6th number is 127
Checking Emma's rule (double previous + 1): 3→7 (3×2+1=7✓), 7→15 (7×2+1=15✓), 15→31 (15×2+1=31✓), 31→63 (31×2+1=63✓). Checking Luis's rule (2n+1 - 1): n=1: 2²-1=3✓, n=2: 2³-1=7✓, n=3: 2⁴-1=15✓, n=4: 2⁵-1=31✓, n=5: 2⁶-1=63✓. Both rules work! For the 6th number: Emma's way: 63×2+1=127. Luis's way: 2⁷-1=128-1=127. Both give 127. Choice C is wrong because the 6th number is 127, not 125. - In a number pattern, each number is 4 more than the number before it. The 3rd number is 15 and the 6th number is 27. What is the 1st number in the pattern?
Answer: The 1st number is 7 in this pattern
If each number is 4 more than the previous, and the 3rd number is 15, then the 2nd number is 15-4=11, and the 1st number is 11-4=7. We can verify: 1st=7, 2nd=11, 3rd=15, 4th=19, 5th=23, 6th=27 ✓. Choice A (5) would give 3rd number as 13, not 15. Choice C (9) would give 3rd number as 17. Choice D (11) would give 3rd number as 19.
- Alex discovers that 10-1=9, 20-2=18, 30-3=27, 40-4=36. He predicts the next two results will be 45 and 54. Is Alex's prediction correct and what is the pattern in the results?
Answer: Alex is correct, and each result is 9 times the tens digit
Following Alex's pattern: 50-5=45 and 60-6=54, so his predictions are correct. Looking at the pattern in results: 9, 18, 27, 36, 45, 54. Each result is indeed 9 times the tens digit (9=9×1, 18=9×2, 27=9×3, 36=9×4, 45=9×5, 54=9×6). The results also increase by 9 each time, but choice A is more precise about the underlying relationship. Choice B is wrong because Alex is correct. Choices C and D focus on the arithmetic progression but miss the deeper pattern.
- Based on the graph shown: What number should replace the question mark?
Answer: 44
The bars show values that increase by 2 each time: 38 → 40 → 42, so the next value is 42 + 2 = 44. Choices A, B, and D do not follow the +2 pattern.
