Triangular numbers, square numbers and cube numbers
Number
- Foundation
- Non-calculator
- 8 questions
- 29 marks
These are original practice questions, not a past paper. This site is independent and not affiliated with AQA or Pearson.
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Write down the first five triangular numbers.
2 marks
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Which of $16$, $20$, $25$ and $30$ are square numbers?
2 marks
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Work out $1^{3}$, $2^{3}$, $3^{3}$, $4^{3}$ and $5^{3}$.
3 marks
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The $n$th triangular number is $\dfrac{n(n + 1)}{2}$. Find the $15$th triangular number.
3 marks
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Find two square numbers, both greater than $0$, that add up to $100$.
4 marks
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Find the number between $50$ and $200$ that is both a square number and a cube number.
4 marks
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The difference between two consecutive triangular numbers is $20$. Find both numbers.
5 marks
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Find the $8$th triangular number and explain why it is also a square number. The sum of the first $n$ cube numbers is the square of the $n$th triangular number. Use this to find $1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3}$.
6 marks
Mark scheme
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Answer. $1, 3, 6, 10, 15$
- Add $2$, then $3$, then $4$, then $5$.
- $1, 3, 6, 10, 15$
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Answer. $16$ and $25$
- $4^{2} = 16$ and $5^{2} = 25$.
- $20$ and $30$ are not squares.
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Answer. $1, 8, 27, 64, 125$
- $1, 8, 27, 64$
- $5^{3} = 125$
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Answer. $120$
- $\dfrac{15 \times 16}{2}$
- $120$
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Answer. $36$ and $64$
- $6^{2} = 36$ and $8^{2} = 64$.
- $36 + 64 = 100$.
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Answer. $64$
- Cube numbers in range: $64$ and $125$.
- $64 = 8^{2} = 4^{3}$.
- $125$ is not a square.
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Answer. $190$ and $210$
- The difference between the $(n - 1)$th and $n$th triangular numbers is $n$.
- So $n = 20$.
- $T_{19} = 190$ and $T_{20} = 210$.
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Answer. The $8$th triangular number is $36 = 6^{2}$. The sum of the cubes is $225$.
- $T_{8} = \dfrac{8 \times 9}{2} = 36$.
- $36 = 6^{2}$, so it is square.
- $T_{5} = 15$, and $15^{2} = 225$.
- Check: $1 + 8 + 27 + 64 + 125 = 225$.