Question - Pattern Expansion and Perimeter Calculation
Solution:
This question involves determining the perimeter of a pattern that is expanding according to a rule. Each new figure in the sequence is formed by adding rectangles to the previous figure, and these additions are highlighted in red in the image.We're given that the perimeters of the first four figures in the sequence are 4 cm, 8 cm, 20 cm, and 56 cm respectively.To find the perimeter of the "Figure 6", we need to identify the pattern by which the perimeters are increasing from one figure to the next:- From Figure 1 to Figure 2, the perimeter increases by 4 cm.- From Figure 2 to Figure 3, the perimeter increases by 12 cm (8 cm + 4 = 12 cm; 8 + 12 = 20).- From Figure 3 to Figure 4, the perimeter increases by 36 cm (20 cm + 3*12 = 56 cm).We can observe that the increment itself is multiplied by 3 each time:- Increment from Figure 1 to 2 is 4 cm.- Increment from Figure 2 to 3 is 12 cm (3 times the previous increment of 4 cm).- Increment from Figure 3 to 4 is 36 cm (3 times the previous increment of 12 cm).Using this pattern, we can calculate the increment from Figure 4 to Figure 5:- Increment from Figure 4 to 5 is 108 cm (3 times the previous increment of 36 cm).Now, let's add this increment to the perimeter of Figure 4:- Perimeter of Figure 5 = 56 cm + 108 cm = 164 cm.Now, we need to calculate the increment for the next figure, which will be 3 times 108 cm:- Increment from Figure 5 to 6 is 324 cm (3 times the previous increment of 108 cm).Finally, let's add this increment to the perimeter of Figure 5 to get the perimeter of Figure 6:- Perimeter of Figure 6 = 164 cm + 324 cm = 488 cm.Therefore, the perimeter of Figure 6 is 488 cm.
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