Question 6 of 29advanced🔍 AnalyzeFill in the Blanks2 marks

In the following pattern, fill in the missing numbers: 1² + 2² + 2² = 3²; 2² + 3² + 6² = 7²; 3² + 4² + 12² = 13²; 4² + 5² + 20² = ()²; 9² + 10² + ()² = (___)²

Correct Answer

4² + 5² + 20² = (21)²; 9² + 10² + (90)² = (91)². The pattern shows that n² + (n+1)² + [n×(n+1)]² = (n² + n + 1)². Simplifying: the result equals (n² + n + 1)². For n = 4: result = 21. For n = 9: third term = 90, result = 91.

Exercise: Figure it Out (Section 1) | Q: 8 | (Page Page 10-11)
For More Understanding

Explanation

The chapter presents a pattern where the sum of three squares equals another square. Observing the pattern: for n = 1: 1² + 2² + 2² = 9 = 3²; for n = 2: 2² + 3² + 6² = 49 = 7². The third term is n×(n+1), and the result is n² + n + 1.

Solution Steps

  1. Step 1: Identify the pattern: third term = n × (n+1)

  2. Step 2: For n = 4: third term = 4×54 \times 5 = 20

  3. Step 3: Result = 4² + 5² + 20² = 16 + 25 + 400 = 441 = 21²

  4. Step 4: For n = 9: third term = 9×109 \times 10 = 90

  5. Step 5: Result = 9² + 10² + 90² = 81 + 100 + 8100 = 8281 = 91²

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