Pattern Rules
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ISEE Middle Level: Quantitative Reasoning › Pattern Rules
Each day, a scientist observes that a certain bacterial culture consumes one-third of the nutrients in its petri dish. If the dish started with a full supply of nutrients on Monday morning, what fraction of the original nutrients will remain at the start of business on Thursday morning?
1/81
8/27
1/3
2/3
Explanation
If the culture consumes one-third of the nutrients, then two-thirds (2/3) of the nutrients remain each day. We need to find the amount remaining on Thursday morning.
- Monday morning: 1 (full supply)
- Tuesday morning: 2/3 of the supply remains from Monday.
- Wednesday morning: 2/3 of Tuesday's amount remains, which is (2/3) * (2/3) = 4/9 of the original.
- Thursday morning: 2/3 of Wednesday's amount remains, which is (2/3) * (4/9) = 8/27 of the original. So, 8/27 of the original nutrients will remain on Thursday morning.
The units digits of the powers of 7 follow a repeating pattern: 7¹=7, 7²=49, 7³=343, 7⁴=2401, 7⁵=16807. The pattern of the units digits is 7, 9, 3, 1, and then it repeats. What is the units digit of 7³⁰?
1
3
7
9
Explanation
The pattern of the units digits of powers of 7 is (7, 9, 3, 1). This is a cycle of length 4. To find the units digit of 7³⁰, we need to determine where in the cycle the 30th power falls. We can do this by dividing the exponent, 30, by the length of the cycle, 4. 30 ÷ 4 = 7 with a remainder of 2. The remainder tells us the position in the cycle.
- A remainder of 1 corresponds to the first digit (7).
- A remainder of 2 corresponds to the second digit (9).
- A remainder of 3 corresponds to the third digit (3).
- A remainder of 0 corresponds to the fourth digit (1). Since the remainder is 2, the units digit of 7³⁰ is the second digit in the pattern, which is 9.
A sequence of numbers starts with 1, 2, 4, 7, 11, 16, ... . The pattern continues. What is the rule to find the next number in the sequence?
Double the previous number and subtract an increasing amount.
Add the term's position number to the previous term.
Add the two previous numbers.
Add a number that is one greater than the number added previously.
Explanation
Let's examine the difference between consecutive terms:
- From 1 to 2, we add 1.
- From 2 to 4, we add 2.
- From 4 to 7, we add 3.
- From 7 to 11, we add 4.
- From 11 to 16, we add 5. The rule is to add a consecutive integer to each term to get the next term. This is described by choice C: "Add a number that is one greater than the number added previously."
A sequence of fractions is 1/3, 2/5, 3/7, 4/9, ... . If the pattern continues, which of the following describes the rule for finding the nth fraction in the sequence?
The numerator is n, and the denominator is the numerator plus 2n-1.
The numerator is n, and the denominator is n + 2.
The numerator is n, and the denominator is 2n + 1.
The numerator is n, and the denominator is the previous denominator plus 2.
Explanation
Let's analyze the numerator and denominator separately in relation to the term number, n.
- Numerators: 1, 2, 3, 4, ... For the nth term, the numerator is simply n.
- Denominators: 3, 5, 7, 9, ... This is an arithmetic sequence. Let's see if we can relate it to n. For n=1, denominator is 3. 2(1)+1 = 3. For n=2, denominator is 5. 2(2)+1 = 5. For n=3, denominator is 7. 2(3)+1 = 7. For n=4, denominator is 9. 2(4)+1 = 9. The rule for the denominator is 2n + 1. Therefore, the rule for the nth fraction is n / (2n + 1).
The terms in a sequence are generated by the rule V = n³ - 2n², where V is the value of the term and n is its position number (n=1, 2, 3...). What is the first term to have a value greater than 50?
The 4th term
The 6th term
The 5th term
The 7th term
Explanation
We need to test values of n starting from 1 until the value of the term, V, is greater than 50.
- For n=1: V = 1³ - 2(1²) = 1 - 2 = -1
- For n=2: V = 2³ - 2(2²) = 8 - 8 = 0
- For n=3: V = 3³ - 2(3²) = 27 - 18 = 9
- For n=4: V = 4³ - 2(4²) = 64 - 32 = 32
- For n=5: V = 5³ - 2(5²) = 125 - 50 = 75 Since 75 is the first value in the sequence greater than 50, the 5th term is the correct answer.
The first five terms of a sequence are 2, 5, 10, 17, 26. Which rule describes the relationship between a term's value (V) and its position number (n) in the sequence?
V = 2n + (n - 2)
Add the next consecutive odd number, starting with 3, to the previous term's value.
V = 3n - 1
V = n² + 1
Explanation
The question asks for an explicit rule that connects the value of a term (V) directly to its position number (n). Let's test the terms:
- For n=1, V=2
- For n=2, V=5
- For n=3, V=10
- For n=4, V=17
- For n=5, V=26 The pattern is that each term is one more than the square of its position number. For example, for n=3, 3² + 1 = 9 + 1 = 10. For n=5, 5² + 1 = 25 + 1 = 26. So, the rule is V = n² + 1.
A pattern starts with 100 and decreases by 7 at each step (Pattern X). Another pattern starts with 5 and doubles at each step (Pattern Y).
Column A: The value of the 6th term in Pattern X.
Column B: The value of the 6th term in Pattern Y.
The two quantities are equal.
The quantity in Column B is greater.
The quantity in Column A is greater.
The relationship cannot be determined from the information given.
Explanation
To find the value in each column, we list the terms of each pattern.
Column A (Pattern X): The sequence is 100, 93, 86, 79, 72, 65, ... The 6th term is 65.
Column B (Pattern Y): The sequence is 5, 10, 20, 40, 80, 160, ... The 6th term is 160.
Comparing the two values, 160 is greater than 65. Therefore, the quantity in Column B is greater.
A sequence of numbers follows the rule: "multiply the previous term by 2, then add 3." The 4th term of the sequence is 37. What is the first term?
2
3
5
7
Explanation
To find the first term, we must work backward from the 4th term. The rule is T_n = 2 * T_(n-1) + 3. The inverse operation is to subtract 3, then divide by 2.
- 4th term = 37
- 3rd term = (37 - 3) / 2 = 34 / 2 = 17
- 2nd term = (17 - 3) / 2 = 14 / 2 = 7
- 1st term = (7 - 3) / 2 = 4 / 2 = 2 The first term is 2. We can check by working forward: 2 -> (22+3)=7 -> (72+3)=17 -> (17*2+3)=37. This confirms the answer.
A pattern is defined by the rule "multiply the previous term by x, then add 2." The first term is 4. The variable x is a positive integer.
Column A: The value of the third term if x = 3.
Column B: 50.
The relationship cannot be determined from the information given.
The quantity in Column A is greater.
The quantity in Column B is greater.
The two quantities are equal.
Explanation
Let's calculate the value for Column A. The first term is 4 and the rule is T_n = x * T_(n-1) + 2. We are given x = 3.
- First term: 4
- Second term: (4 * 3) + 2 = 12 + 2 = 14
- Third term: (14 * 3) + 2 = 42 + 2 = 44 So, the quantity in Column A is 44. The quantity in Column B is 50. Since 50 is greater than 44, the quantity in Column B is greater.
Leo is building pyramids with blocks. A Level 1 pyramid is a single block. For each subsequent level, he builds a square base and places the previous level's pyramid on top. A Level 2 pyramid has a 2x2 base, and a Level 3 pyramid has a 3x3 base. The total number of blocks in a pyramid is the sum of the blocks in its base and all the blocks in the levels above it. Following this pattern, how many total blocks are in a Level 4 pyramid?
16
21
30
64
Explanation
Let's determine the total number of blocks for each level:
- Level 1: 1² = 1 block total.
- Level 2: Has a 2x2 base (4 blocks) plus the Level 1 pyramid on top (1 block). Total = 4 + 1 = 5 blocks. (This is 1² + 2²).
- Level 3: Has a 3x3 base (9 blocks) plus the Level 2 pyramid on top (5 blocks). Total = 9 + 5 = 14 blocks. (This is 1² + 2² + 3²). The pattern is that the total number of blocks in a Level n pyramid is the sum of the squares from 1 to n. For a Level 4 pyramid, the total is 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30 blocks.