- Sequence S starts with 1 and is generated by the rule "add 8 to the previous term." Sequence T starts with 1 and is generated by the rule "multiply the previous term by 2."
Column A: The amount that is added to the 99th term to get the 100th term in Sequence S.
Column B: The amount that is added to the 7th term to get the 8th term in Sequence T.
Answer: The quantity in Column B is greater.
Column A asks for the difference between the 100th and 99th terms of Sequence S. Sequence S is an arithmetic sequence where 8 is always added. So, the amount added is always 8.
Column B asks for the difference between the 8th and 7th terms of Sequence T. Sequence T is a geometric sequence.
- 7th term: T₇ = 1 * 2⁷⁻¹ = 2⁶ = 64.
- 8th term: T₈ = 1 * 2⁸⁻¹ = 2⁷ = 128.
The amount added is T₈ - T₇ = 128 - 64 = 64.
Comparing the two quantities, 8 (Column A) is less than 64 (Column B).
- The number of diagonals in a polygon follows a pattern. A quadrilateral (4 sides) has 2 diagonals. A pentagon (5 sides) has 5 diagonals. A hexagon (6 sides) has 9 diagonals. A heptagon (7 sides) has 14 diagonals. Based on this pattern, how many diagonals does a nonagon (9 sides) have?
Answer: 27
Let's analyze the pattern of increase in the number of diagonals.
- From 4 sides to 5 sides, the diagonals increase from 2 to 5 (an increase of 3).
- From 5 sides to 6 sides, the diagonals increase from 5 to 9 (an increase of 4).
- From 6 sides to 7 sides, the diagonals increase from 9 to 14 (an increase of 5).
The pattern is that for each additional side, the number of diagonals increases by an amount that is one greater than the previous increase.
- For an octagon (8 sides), the increase would be 6. Diagonals = 14 + 6 = 20.
- For a nonagon (9 sides), the increase would be 7. Diagonals = 20 + 7 = 27.
- 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?
Answer: The numerator is n, and the denominator is 2n + 1.
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).
- Two alarms beep at regular intervals. Alarm A beeps every 6 minutes. Alarm B beeps every 10 minutes. They both beep together at 1:00 PM. What is the first time after 2:00 PM that they will beep together again?
Answer: 2:30 PM
To find when the alarms beep together, we need to find the least common multiple (LCM) of their intervals, 6 minutes and 10 minutes.
- Multiples of 6: 6, 12, 18, 24, 30, 36, ...
- Multiples of 10: 10, 20, 30, 40, ...
The LCM of 6 and 10 is 30. So, the alarms beep together every 30 minutes.
They start at 1:00 PM. The times they beep together are:
- 1:00 PM
- 1:30 PM
- 2:00 PM
- 2:30 PM
- 3:00 PM
The question asks for the first time after 2:00 PM. The next time on the list after 2:00 PM is 2:30 PM.
- A manufacturing machine has a rule for converting an input value, x, into an output value, y. The rule is: "multiply the input by -4, then subtract the result from 10." Which equation correctly represents this rule?
Answer: y = 10 - (-4x)
Let's break down the rule into algebraic steps.
1. "multiply the input by -4": This gives -4x.
2. "subtract the result from 10": This means we start with 10 and subtract the result from step 1. So, it's 10 - (-4x).
Combining these, we get the equation y = 10 - (-4x), which simplifies to y = 10 + 4x. Choice C matches the unsimplified, direct translation of the rule.
- Pattern A follows the rule Tₙ = 2n + k, where k is a constant. The 5th term of Pattern A is 13.
Pattern B follows the rule Tₙ = 4n + j, where j is a constant. The 3rd term of Pattern B is 11.
Column A: The value of the 10th term of Pattern A.
Column B: The value of the 6th term of Pattern B.
Answer: The two quantities are equal.
First, determine the complete rule for each pattern.
For Pattern A: Tₙ = 2n + k. Given T₅ = 13, we have 13 = 2(5) + k, which gives 13 = 10 + k, so k = 3. The rule for Pattern A is Tₙ = 2n + 3.
For Pattern B: Tₙ = 4n + j. Given T₃ = 11, we have 11 = 4(3) + j, which gives 11 = 12 + j, so j = -1. The rule for Pattern B is Tₙ = 4n - 1.
Now calculate the required terms:
Column A: The 10th term of Pattern A is T₁₀ = 2(10) + 3 = 23.
Column B: The 6th term of Pattern B is T₆ = 4(6) - 1 = 23.
The two quantities are equal.
- A car rental company charges a flat fee plus a cost per mile. A 100-mile trip costs $80. A 150-mile trip costs $95. What is the rule used to calculate the total cost (C) for a trip of m miles?
Answer: C = 0.30m + 50
This problem describes a linear relationship. First, find the cost per mile. The difference in cost for the two trips is $95 - $80 = $15. The difference in miles is 150 - 100 = 50 miles. So, the cost per mile is $15 / 50 miles = $0.30 per mile.
Now, find the flat fee. Let the flat fee be F. The total cost is C = 0.30m + F. Using the 100-mile trip:
$80 = 0.30(100) + F
$80 = $30 + F
F = $50.
The rule is C = 0.30m + 50. We can check this with the 150-mile trip: C = 0.30(150) + 50 = 45 + 50 = $95. This is correct.
- 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?
Answer: 9
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.
- 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?
Answer: 8/27
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.
- 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.
Answer: The quantity in Column B is greater.
Let's calculate the value for Column A. The first term is 4 and the rule is Tn = 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?
Answer: 30
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.
- 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?
Answer: The 5th term
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.
- A numerical sequence follows an alternating two-step rule. The sequence begins 2, 6, 3, 9, 6, 18, 15, ... If the pattern continues, what is the next number in the sequence?
Answer: 45
Let's analyze the steps between consecutive terms:
- From 2 to 6: Multiply by 3.
- From 6 to 3: Subtract 3.
- From 3 to 9: Multiply by 3.
- From 9 to 6: Subtract 3.
- From 6 to 18: Multiply by 3.
- From 18 to 15: Subtract 3.
The pattern is to alternately multiply by 3 and subtract 3. The last operation was subtracting 3 to get 15. The next operation must be to multiply by 3. So, the next term is 15 * 3 = 45.
- 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?
Answer: V = n² + 1
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 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?
Answer: 2
To find the first term, we must work backward from the 4th term. The rule is Tn = 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 -> (2*2+3)=7 -> (7*2+3)=17 -> (17*2+3)=37. This confirms the answer. - 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?
Answer: Add a number that is one greater than the number added previously.
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 is defined by the rule that each term after the second is the sum of the two preceding terms.
Column A: The 8th term if the first two terms are 1 and 4.
Column B: The 7th term if the first two terms are 2 and 5.
Answer: The quantity in Column A is greater.
We need to generate the terms for each sequence.
For Column A, the sequence starts with 1, 4. The subsequent terms are the sum of the previous two:
1, 4, 5, 9, 14, 23, 37, 60. The 8th term is 60.
For Column B, the sequence starts with 2, 5:
2, 5, 7, 12, 19, 31, 50. The 7th term is 50.
Comparing the quantities, 60 (Column A) is greater than 50 (Column B).
- 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.
Answer: The quantity in Column B is greater.
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 pattern is described by the rule V = -5n + 32, where V is the value of the term and n is its position number (1, 2, 3, ...). What is the position number of the first term in the pattern to have a negative value?
Answer: The 7th term
We need to find the smallest integer n for which the value V is less than 0.
-5n + 32 < 0
Add 5n to both sides:
32 < 5n
Divide by 5:
32/5 < n
6.4 < n
Since n must be an integer representing the position number, the smallest integer n that is greater than 6.4 is 7. So, the 7th term is the first to have a negative value.
Let's check: For n=6, V = -5(6) + 32 = -30 + 32 = 2. For n=7, V = -5(7) + 32 = -35 + 32 = -3.
