Question 5 of 19intermediate🔧 ApplyNumerical2 marks

What would you call the following sequence of numbers? 1 7 19 37 That's right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence?

Correct Answer

The next number in the hexagonal sequence is 61.

Formula: H(n)=3n(n−1)+1H(n) = 3n(n-1) + 1

Verification:

H(1)=3×1×0+1=1H(1) = 3×1×0 + 1 = 1 H(2)=3×2×1+1=7H(2) = 3×2×1 + 1 = 7 H(3)=3×3×2+1=19H(3) = 3×3×2 + 1 = 19 H(4)=3×4×3+1=37H(4) = 3×4×3 + 1 = 37 H(5)=3×5×4+1=61H(5) = 3×5×4 + 1 = 61

Alternative Method (Pattern of Differences): Differences between consecutive terms: 7 - 1 = 6 19 - 7 = 12 37 - 19 = 18

The differences increase by 6 each time: 6, 12, 18, ...

Next difference: 18 + 6 = 24

Next number: 37 + 24 = 61

Exercise: Figure it Out (Section 3) | Q: 4 | (Page Page 5)
For More Understanding

Explanation

Hexagonal numbers follow the pattern where each term represents dots that can form a hexagon shape. The formula for the nth hexagonal number is H(n) = 3n(n-1) + 1. Checking: H(1) = 3×13 \times 1×0 + 1 = 1; H(2) = 3×23 \times 2×1 + 1 = 7; H(3) = 3×33 \times 3×2 + 1 = 19; H(4) = 3×43 \times 4×3 + 1 = 37; H(5) = 3×53 \times 5×4 + 1 = 61. Alternatively, the differences between consecutive terms are: 7-1=6, 19-7=12, 37-19=18, and the next difference would be 24 (increasing by 6 each time), so 37+24=61.

Solution Steps

  1. Step 1: Identify the pattern in differences: 7-1=6, 19-7=12, 37-19=18

  2. Step 2: Notice that differences increase by 6 each time: 6, 12, 18, ...

  3. Step 3: The next difference is 18+6=24

  4. Step 4: Calculate the next number: 37+24=61

PreviousNext