Class 8 Patterns in Cubes — practice questions with answers
Cubes and Roots · 5 sample questions from the school syllabus. Try each one, then open the answer and the worked solution.
1. Cubes can be built by adding blocks of consecutive odd numbers: 1 = 1³, then 3 + 5 = 8 = 2³, then 7 + 9 + 11 = 27 = 3³. The next block 13 + 15 + 17 + 19 = 64 is the cube of which number?
A4
B3
C5
D8
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Answer: A) 4
64 = 4³, so it is the cube of 4. Each cube n³ is the sum of the next n consecutive odd numbers (here 4 odd numbers give 4³).
2. Find the value of 10³ - 9³.
A271
B542
C246
D296
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Answer: A) 271
10³ - 9³ = 1000 - 729 = 271.
3. Meera notices that the gaps between consecutive cubes form a pattern: 2³ - 1³ = 7 3³ - 2³ = 19 4³ - 3³ = 37 These differences increase by 12, then 18, ... What is the difference 5³ - 4³?
4. A florist in Pune arranges marigold pots in a triangular-number then cube pattern. She finds that 1³ + 2³ + 3³ + 4³ + 5³ equals the square of the 5th triangular number. The 5th triangular number is 1+2+3+4+5 = 15. What is the total number of pots?
A245
B450
C225
D205
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Answer: C) 225
Sum of first 5 cubes = 15² = 225 pots.
5. The sum of the first n cubes is a perfect square for every n, since 1³+...+n³ = (2n(n+1))2. For which value of n is this sum equal to 3025?
A9
B11
C20
D10
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Answer: D) 10
We need (2n(n+1))2=3025, so 2n(n+1)=55, giving n(n+1) = 110 = 10 x 11. Thus n = 10. Check: (210×11)2=552=3025.