Question 15 of 51intermediate🔧 ApplyEssay2 marks

Can you predict the number of squares in Stages 5 and 6 of the sequence? In Stages 10, 11 and 12? In Stage 20? At any stage?

Correct Answer

Yes, we can predict the number of squares. The sequence is 1, 5, 9, 13. We observe that at each stage, 4 squares are added to the corners. The number of squares at successive stages is 1, 1+4, 1+4+4, 1+4+4+4, and so on.

Using this, for Stage 5, the count is 1 + 4×44 \times 4 = 17, and for Stage 6, it is 1 + 4×54 \times 5 = 21. For Stages 10, 11, and 12, the number of squares will be 37, 41, and 45 respectively. For Stage 20, it will be 1 + 4×194 \times 19 = 77.

At any stage n, the number of squares is given by the general formula 1 + 4(n - 1), which simplifies to 4n - 3.

Exercise: Think and Reflect | Q: 1 | (Page 7)
For More Understanding

Explanation

The textbook explains that the sequence of squares follows the pattern 1, 1+4, 1+4+4, 1+4+4+4, etc., corresponding to 1, 5, 9, 13. This means 4 squares are added at each successive stage. To find the number of squares at any stage n, we start with 1 square and add 4 squares for each of the (n-1) subsequent stages, leading to the formula 1 + 4(n-1).

Solution Steps

  1. Step 1: Identify the pattern from the sequence 1, 5, 9, 13, where 4 squares are added at each stage.

  2. Step 2: Calculate for Stages 5 and 6: 1 + 4(4) = 17; 1 + 4(5) = 21.

  3. Step 3: Calculate for Stages 10, 11, 12: 1 + 4(9) = 37; 1 + 4(10) = 41; 1 + 4(11) = 45.

  4. Step 4: Calculate for Stage 20: 1 + 4(19) = 77.

  5. Step 5: Generalize for any stage n: 1 + 4(n-1) = 4n - 3.

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