Question 13 of 19advanced🔧 ApplyShort Answer3 marks

What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Can you explain it with a picture?

Correct Answer

Triangular numbers: 1, 3, 6, 10, 15, 21, ...

Calculation: 1×61 \times 6 + 1 = 7 3×63 \times 6 + 1 = 19 6×66 \times 6 + 1 = 37 10×610 \times 6 + 1 = 61 15×615 \times 6 + 1 = 91

Resulting sequence: 7, 19, 37, 61, 91, ...

Answer: These are the hexagonal numbers starting from the second term.

Hexagonal numbers: 1, 7, 19, 37, 61, 91, ...

Formula relationship: H(n) = 3n(n − 1) + 1 T(n − 1) = (n − 1)n/2 ∴ 6 × T(n − 1) + 1 = 3n(n − 1) + 1 = H(n)

Pictorial explanation: Hexagonal numbers can be visualized as dots arranged in hexagon shapes. The relationship T(n) × 6 + 1 = H(n + 1) shows that hexagonal arrangements can be built from triangular arrangements with a central dot.

Exercise: Figure it Out (Section 4) | Q: 7 | (Page Page 8)
For More Understanding

Explanation

The triangular numbers are 1, 3, 6, 10, 15, 21, ... Multiplying by 6 and adding 1: 7, 19, 37, 61, 91, 127, ... These match hexagonal numbers starting from the second hexagonal number. The hexagonal numbers are 1, 7, 19, 37, 61, 91, ... The formula H(n) = 3n(n-1) + 1 relates to triangular numbers since T(n-1) = (n-1)n/2, so 6×T(n-1) + 1 = 3n(n-1) + 1 = H(n).

Solution Steps

  1. Identify triangular numbers: 1, 3, 6, 10, 15, 21, ...

  2. Multiply each triangular number by 6 and add 1

  3. Observe the resulting sequence: 7, 19, 37, 61, 91, ...

  4. Identify the sequence as hexagonal numbers starting from the second term

  5. Verify using formula: H(n) = 3n(n − 1) + 1 = 6 × T(n − 1) + 1

  6. Explain pictorially: Hexagonal arrangements built from triangular arrangements with a central dot

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