2, 9, 28, 65,…217
Answer: D
Correct answer: 126
Each term is one more than a cube: \( 1^3+1=2 \), \( 2^3+1=9 \), \( 3^3+1=28 \), \( 4^3+1=65 \), so the next is \( 5^3+1 = \mathbf{126} \), followed by \( 6^3+1 = 217 \).
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110, 132, 156, 182,….
Answer: D
Correct answer: 210
Each term is the product of two consecutive numbers: \( 10\cdot11=110 \), \( 11\cdot12=132 \), \( 12\cdot13=156 \), \( 13\cdot14=182 \), so the next is \( 14\cdot15 = \mathbf{210} \).
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0, 2, 6, 12, 20,….42
Answer: B
Correct answer: 30
Each term is the product of a number and the one below it: \( 1\cdot0=0 \), \( 2\cdot1=2 \), \( 3\cdot2=6 \), \( 4\cdot3=12 \), \( 5\cdot4=20 \), so the next is \( 6\cdot5 = \mathbf{30} \), followed by \( 7\cdot6 = 42 \).
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1001, 8001, 27001, 64001, 125001,…
Answer: B
Correct answer: 216001
Each term is a cube multiplied by a thousand, plus one: \( 1000\cdot1+1 \), \( 1000\cdot8+1 \), \( 1000\cdot27+1 \), \( 1000\cdot64+1 \), \( 1000\cdot125+1 \), so the next is \( 1000 \times 216 + 1 = \mathbf{216001} \).
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1, 8, 27, 64,…. 216
Answer: A
Correct answer: 125
The terms are the cubes: \( 1^3, 2^3, 3^3, 4^3 \), so the next is \( 5^3 = \mathbf{125} \), followed by \( 6^3 = 216 \).
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Which number would replace the question mark (?) in the series 2, 7, 14, 23, ?, 47
Answer: B
Correct answer: 34
The gaps are the successive odd numbers: \( 7-2=5 \), \( 14-7=7 \), \( 23-14=9 \). The next gap is 11, giving \[ 23 + 11 = \mathbf{34} \] and \( 34 + 13 = 47 \), as shown.
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99, 80, 63,….35
Answer: B
Correct answer: 48
Each term is one less than a square, counting downwards: \( 10^2-1=99 \), \( 9^2-1=80 \), \( 8^2-1=63 \), so the next is \( 7^2-1 = \mathbf{48} \), followed by \( 6^2-1 = 35 \).
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15, 31, 63, 127, 255, --
Answer: B
Correct answer: 511
Each term is one less than a power of two: \( 2^4-1=15 \), \( 2^5-1=31 \), \( 2^6-1=63 \), \( 2^7-1=127 \), \( 2^8-1=255 \), so the next is \( 2^9-1 = \mathbf{511} \). Equivalently, each term is twice the one before, plus one.
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1000, 8000, 27000, 64000,….
Answer: B
Correct answer: 125000
Each term is a cube multiplied by a thousand: \( 1^3, 2^3, 3^3, 4^3 \) times 1000, so the next is \( 5^3 \times 1000 = \mathbf{125000} \).
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