Question 16 of 19intermediate🔧 ApplyShort Answer3 marks

Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?

Correct Answer

Counting lines in Complete Graphs: K₂ has 1 line K₃ has 3 lines K₄ has 6 lines K₅ has 10 lines K₆ has 15 lines

Sequence obtained: 1, 3, 6, 10, 15, ... This is the Triangular numbers sequence.

Explanation: In a complete graph K_n with n points, each point is connected to every other point. The number of lines equals the number of ways to choose 2 points from n points. Formula: n(n-1)/2

Verification: K₂: 2×12 \times 1/2 = 1 K₃: 3×23 \times 2/2 = 3 K₄: 4×34 \times 3/2 = 6 K₅: 5×45 \times 4/2 = 10 K₆: 6×56 \times 5/2 = 15

∴ The values 1, 3, 6, 10, 15 are exactly the triangular numbers.

Exercise: Figure it Out (Section 6) | Q: 2 | (Page Page 11-12)
For More Understanding

Explanation

The chapter shows Complete Graphs K₂ through K₆. In a complete graph, every vertex is connected to every other vertex by exactly one edge. The number of edges in K_n is the number of ways to choose 2 vertices from n vertices, which equals n(n-1)/2. This formula generates triangular numbers: 1, 3, 6, 10, 15, ...

Solution Steps

  1. Count the number of lines in each complete graph from K₂ through K₆.

  2. List the values obtained: 1, 3, 6, 10, 15.

  3. Identify the sequence as the Triangular numbers sequence.

  4. Apply the formula n(n-1)/2 to calculate the number of lines in K_n.

  5. Verify that the formula produces triangular numbers for each graph.

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