Problem 70 - Single Variable Calculus

Question

An infinite sequence of nested squares is constructed as follows: Starting with a square with a side of length 2 , each square in the sequence is constructed from the preceding square by drawing line segments connecting the midpoints of the sides of the square. Find the sum of the areas of all the squares in the sequence.

Step-by-Step Solution

Answer

The sum of the areas of all the squares in the infinite sequence of nested squares is 8. This is computed by recognizing the sequence as an infinite geometric series with the first term \(a_1 = 4\) and the common ratio \(r = \frac{1}{2}\), and using the formula \(A = \frac{a_1}{1 - r} = \frac{4}{1 - \frac{1}{2}} = 8\).

Step 1: Identify the relationship between consecutive square side lengths

For this step, let's first look at how to construct the next square in the sequence. Given a square with side length s, we draw a line segment connecting the midpoints of each side of the square, which will create a smaller square inside. Let's call the side length of this new square s'. Since the new square is made up of right triangles from the original square, where each leg of the triangle measures s/2, we can use the Pythagorean theorem to compute the side length of the new square, s'. \[s'^2 = \left(\frac{s}{2}\right)^2 + \left(\frac{s}{2}\right)^2\]

Step 2: Compute the side length of the next square in the sequence

Using the relationship found in Step 1, we can now compute the side length of the next square in the sequence. \[s'^2 = \left(\frac{s}{2}\right)^2 + \left(\frac{s}{2}\right)^2 = \frac{s^2}{2}\] Taking the square root of both sides, we have: \[s' = \frac{s}{\sqrt{2}}\]

Step 3: Find the area of each square in the sequence

We will now find the area of each square in the sequence using the side lengths. For the first square, with side length 2, the area is: \[A_1 = (2)^2 = 4\] Now, using the relationship found in Step 2, we can find the side length and the area of the next square in the sequence: \[s' = \frac{s}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}\] and the area is: \[A_2 = (\sqrt{2})^2 = 2\] We can continue this process of finding the side lengths and areas of the subsequent squares in the sequence.

Step 4: Sum the areas of all squares in the sequence

Since we are dealing with an infinite sequence of nested squares, we can represent the sum of their areas as an infinite geometric series: \[A = A_1 + A_2 + A_3 + \ldots = 4 + 2 + \frac{1}{2} \cdot 2 + \frac{1}{2^2} \cdot 2 + \ldots\] In this geometric series, the first term \(a_1 = 4\), and the common ratio \(r = \frac{1}{2}\). To find the sum of the infinite geometric series, we can use the following formula: \[A = \frac{a_1}{1 - r} \]

Step 5: Compute the sum of the geometric series

Now we can plug the values of \(a_1\) and \(r\) into the formula and find the sum of the areas of all squares: \[A = \frac{4}{1 - \frac{1}{2}} = \frac{4}{\frac{1}{2}} = 8\] Therefore, the sum of the areas of all the squares in the sequence is 8.

Key Concepts

Nested Squares

Imagine starting with a single sheet of paper and folding it in half to create a smaller square within it, then continuing this process indefinitely. This is the concept of nested squares, a fascinating visual arrangement where by folding or, in our exercise, drawing line segments from the midpoints of a square, you create another square nested within the original one.

The resulting figure consists of an initial larger square and a series of progressively smaller squares, each nestled inside the other. In our exercise, this process begins with a square of side length 2 and continues infinitely. Students often visualize nested squares as a mesmerizing, ever-decreasing pattern, captivating in its simplicity and symmetry.

Pythagorean Theorem

A cornerstone of geometry is the Pythagorean theorem, a principle that relates the lengths of the sides of a right-angled triangle. The theorem posits that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In algebraic terms, for a triangle with sides of length 'a', 'b', and 'c', where 'c' is the hypotenuse, the theorem is expressed as:\[\begin{equation}c^2 = a^2 + b^2\text{.}ewline\end{equation}\]This formula is profoundly useful in diverse areas of mathematics and has been a key piece in solving our nested squares exercise. By applying the Pythagorean theorem, we can determine the side length of the new, nested squares.

Convergence of Series

In mathematics, particularly in calculus, the convergence of a series is a concept that explores whether the sum of an infinite sequence of numbers settles or 'converges' to a finite value. If the series does converge, it means that as we add more and more terms, the total sum approaches a specific number. This is crucial for our exercise, as we look at the infinite series of the areas of nested squares.

The idea of convergence is central to the understanding of infinite sequences and series, as it helps to determine if discussing the 'sum' of infinite terms even makes sense. For instance, the sum of the areas of our nested squares will only be meaningful if this series converges to a certain value.

Geometric Sequence

Picture a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. That's a geometric sequence, a chain of numbers that share this recursive, multiplicative relationship. It can be an escalating series where the numbers grow larger, or like with our nested squares, it can dwindle down to fractions of the starting value.

Geometric sequences are inherent to many natural phenomena and mathematical concepts, including financial calculations like compound interest. In the context of our exercise on nested squares, the areas of the squares form a decreasing geometric sequence, directly impacting the method used to find the sum of all areas in the sequence.