The sides of the outside square in the figure are 8 inches long. The corners of the next smaller square are at the midpoints of the outside square, and so on.
a.) find the sum of the perimeters of the first three squares.
Imagine that the process of nesting squares continues forever. The sum of the perimeters approaches a finite number.
b.) What is that number?
clearly the ratio of each smaller square is 1/√2. So, you have a geometric progression with
a = 8
r = 1/√2
Now just apply your formulas for the sum of such a sequence.
a.) To find the sum of the perimeters of the first three squares, we need to calculate the perimeter of each square and then sum them up.
First Square:
The sides of the outside square are 8 inches long, so the perimeter of the first square is 4 * 8 = 32 inches.
Second Square:
The second square is located at the midpoints of the sides of the first square. Since the sides of the first square are 8 inches long, the sides of the second square are half that length, which is 8/2 = 4 inches. Therefore, the perimeter of the second square is 4 * 4 = 16 inches.
Third Square:
Similarly, the sides of the third square are half the length of the second square, which is 4/2 = 2 inches. Therefore, the perimeter of the third square is 4 * 2 = 8 inches.
To find the sum of the perimeters of the first three squares, we add the perimeters together: 32 + 16 + 8 = 56 inches.
b.) The sum of the perimeters of the first three squares is 56 inches. Now, if we imagine extending this nesting of squares indefinitely, we can consider what happens to the sum of the perimeters as the squares become increasingly smaller.
Each subsequent square has sides half the length of the previous square. As we continue to nest the squares, their perimeters will tend towards a limit.
In this case, the limit exists because each time we divide the side length by 2, the new side length becomes closer and closer to 0 (but never quite reaches it). This means that the sum of the perimeters of the infinitely nested squares will approach a finite value.
To find this value, we can use the concept of a geometric series. The sum of an infinite geometric series with a common ratio less than 1 can be determined by using the formula:
Sum = a / (1 - r)
Where "a" is the first term of the series and "r" is the common ratio.
In this case, the first term "a" is 56 inches (from the sum of the first three squares). The common ratio "r" is 1/2, since each subsequent square has sides half the length of the previous square.
Using the formula, we find:
Sum = 56 / (1 - 1/2) = 112 inches.
Therefore, the sum of the perimeters of the infinitely nested squares is 112 inches.