Question 16 of 17intermediate🔍 AnalyzeShort Answer3 marks

Evaluate the following sequence of fractions: 1/3, (1+3)/(5+7), (1+3+5)/(7+9+11). What do you observe? Can you explain why this happens?

Correct Answer

Step 1: First fraction 1/3 = 1/3

Step 2: Second fraction (1+3)/(5+7) = 4/12 = 1/3

Step 3: Third fraction (1+3+5)/(7+9+11) = 9/27 = 1/3

Observation: All three fractions equal 1/3.

Exercise: Figure it Out (Section 3) | Q: 10 | (Page Page 12)
For More Understanding

Explanation

The chapter hints at recalling the sum of first n odd numbers. The sum of first n odd numbers (1+3+5+...+(2n-1)) = n². The next n odd numbers are (2n+1), (2n+3), ..., (4n-1), and their sum is 3n². So the ratio is always 1/3.

Solution Steps

  1. Step 1: 1/3 = 1/3

  2. Step 2: (1+3)/(5+7) = 4/12 = 1/3

  3. Step 3: (1+3+5)/(7+9+11) = 9/27 = 1/3

  4. Step 4: Sum of first n odd numbers = n²

  5. Step 5: Sum of next n odd numbers = 3n²

  6. Step 6: Ratio = n²/3n² = 1/3

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