Number Pattern Rules
Help Questions
ISEE Middle Level: Mathematics Achievement › Number Pattern Rules
A baker decorates a cake with concentric rings of flowers. The innermost ring has 5 flowers. Each successive ring has 4 more flowers than the ring just inside of it. Which rule describes how to find the number of flowers in a ring given the number of flowers in the previous, smaller ring?
Add 5 to the number of flowers in the previous ring.
Multiply the number of flowers in the previous ring by 4.
Multiply the previous ring's flower count by 2 and subtract 6.
Add 4 to the number of flowers in the previous ring.
Explanation
The problem states that each new ring has 4 more flowers than the previous one. This describes an arithmetic sequence with a common difference of 4. The sequence of flowers is 5, 9, 13, 17, ... To get from one term to the next, you simply add 4.
A pattern is formed by starting with 81. Each new number is found by applying the same operation to the previous number. The pattern is 81, 27, 9, 3, ... What is the rule?
Subtract two-thirds of the previous number.
Take the square root of the previous number.
Divide the previous number by 3.
Subtract 54 from the previous number.
Explanation
To get from 81 to 27, you can divide by 3. To get from 27 to 9, you can divide by 3. To get from 9 to 3, you can divide by 3. The rule is to divide the previous number by 3. Choice D would work for the first step (81 - 54 = 27), but fails for subsequent steps (27 - 18 = 9, but 9 - 6 = 3, not the pattern continuation).
In a video game, a player starts with 100 points. For the first level completed, 20 points are added. For each subsequent level, the number of points added is 10 more than the points added for the previous level. What rule determines the player's new score after completing a level?
Add 10 more than the point value of the previous level's reward to the current score.
Add 20 to the player's current score after every level.
Add 10 times the number of the completed level to the current score.
Multiply the player's current score by 1.2 after each level.
Explanation
The sequence of scores is 100, 120, 150, 190, ... The points added are +20, +30, +40, ... Each time, the reward increases by 10. So, to find the new score, you add an amount to the current score that is 10 greater than the amount added for the prior level.
In the sequence 2, 6, 15, 31, 56, ..., the difference between consecutive terms creates its own pattern. What is the rule to find the next term?
Add the next consecutive perfect square, starting with adding 4.
Add the square of the previous term's position number.
Add the two previous terms and subtract 1.
Multiply the previous term by a decreasing factor.
Explanation
First, find the differences between the terms: 6 - 2 = 4. 15 - 6 = 9. 31 - 15 = 16. 56 - 31 = 25. The differences are 4, 9, 16, 25, which are the perfect squares \(2^2, 3^2, 4^2, 5^2\). The rule is to add the next consecutive perfect square.
The following sequence uses a rule that alternates between two different operations: 2, 4, 1, 3, 0, 2, ... What is the rule for this pattern?
Alternately add 2, then divide by 4.
Alternately add 2, then subtract 3.
Alternately subtract 1, then add 2.
Double the previous number, then subtract 3.
Explanation
Check the operations between consecutive terms. 2 to 4 is +2. 4 to 1 is -3. 1 to 3 is +2. 3 to 0 is -3. 0 to 2 is +2. The pattern alternates between adding 2 and subtracting 3.
A sequence of numbers begins 50, 48, 44, 38, 30, ... What is the rule to determine the next number in this sequence?
Subtract the next consecutive even number, starting with 2.
Subtract 2 from the previous number.
Subtract an amount that doubles with each step, starting with 2.
Divide the previous number by 2 and add a changing value.
Explanation
Examine the differences between terms: 50 - 48 = 2. 48 - 44 = 4. 44 - 38 = 6. 38 - 30 = 8. The amounts being subtracted are consecutive even numbers (2, 4, 6, 8). The rule is to subtract the next consecutive even number.
Each number in the pattern 0.5, 0.75, 1.25, 2, 3.25, ... is determined by the numbers that come before it. What rule is used to find the next number in this pattern?
Add the two preceding numbers.
Add 0.25 to the previous number.
Multiply the previous number by 1.5.
Add 0.25, then 0.50, then 0.75, and so on.
Explanation
This pattern follows a Fibonacci-like rule. Check by adding the two previous terms to get the next: 0.5 + 0.75 = 1.25. 0.75 + 1.25 = 2.0. 1.25 + 2.0 = 3.25. The rule is to add the two preceding numbers.
A sequence starts with 4. Each subsequent number is found by applying the same two operations in the same order to the previous number. The sequence is 4, 9, 19, 39, 79, ... What is the rule?
Add a number that doubles each time, starting with 5.
Add 5 to the previous number, then double the result.
Multiply the previous number by 2, then add 1.
Multiply the previous number by 3, then subtract 3.
Explanation
The stem specifies that the same two operations are applied each time. Let's test choice C. For 4 to 9: (4 × 2) + 1 = 9. For 9 to 19: (9 × 2) + 1 = 19. For 19 to 39: (19 × 2) + 1 = 39. For 39 to 79: (39 × 2) + 1 = 79. This rule works for the entire sequence. Choice B also describes the pattern correctly (differences are 5, 10, 20, 40), but it does not use the 'same two operations' each time, as the number being added changes.
How is the sequence 10, 7, 12, 9, 14 generated?
Add 2 each time
Subtract 3 each time
Multiply by 2, then subtract 1
Subtract 3, then add 5, repeat
Explanation
This question tests middle school mathematics skills, specifically identifying rules that govern number patterns. Understanding number patterns involves recognizing the rule that consistently applies to each number in a sequence, such as addition, subtraction, multiplication, or division. In this problem, the sequence 10, 7, 12, 9, 14 follows an alternating pattern: subtract 3 (10-3=7), then add 5 (7+5=12), subtract 3 (12-3=9), then add 5 (9+5=14). The correct choice, 'Subtract 3, then add 5, repeat,' accurately describes this alternating operation pattern. A common distractor, such as 'Add 2 each time,' fails because it doesn't account for the decreasing values at positions 2 and 4. Teaching strategies include having students track whether each step increases or decreases and by how much, creating a visual pattern of operations. Remind students that some sequences use alternating rules rather than a single consistent operation.
A sequence of numbers starts with 3. Each subsequent number is found using the same two-step rule. The first four numbers are 3, 7, 15, and 31. What is the rule used to generate the numbers in this sequence?
Multiply the previous number by 2, then subtract 1.
Multiply the previous number by 2, then add 1.
Multiply the previous number by 3, then subtract 2.
Add 4 to the previous number.
Explanation
To find the rule, check the relationship between consecutive terms. From 3 to 7: (3 × 2) + 1 = 7. From 7 to 15: (7 × 2) + 1 = 15. From 15 to 31: (15 × 2) + 1 = 31. The rule is to multiply the previous number by 2 and then add 1.