Question

1. The sides of a square 1 cm, have lines drawn between them connecting adjoining sides with their midpoints. This creates another square within the original and this process is continued indefinitely. (Given, Pythagoras theorem: Hypotenuse 2 = Perpendicular 2 + Base 2) a. Show the sequence of the sides for first 4 terms. [1] b. Show the sequence for the areas of the squares for first 4 terms. [1] c. Calculate the sum of the areas of the infinite squares.

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Answer 1

The sequence of the areas of the squares is a geometric progression,

and the sum to infinity can be determined by using a formula.

Correct responses:

a. 1,  0.5·√2,  0.5,  0.25·√2

b. 1, 0.5, 0.25, 0.125

c. The sum of the areas of the infinite squares is 2

How the the sequence of the areas of the squares be determined?

Given;

Length of the side of the square = 1 cm.

The inscribed square is drawn by connecting the midpoints.

Pythagoras's theorem is; Hypotenuse² = Perpendicular² + Base²

Leg length of the second (inscribed) square = 1 ÷ 2 = 0.5

Side length of the second square = √(0.5² + 0.5²) = 0.5·√2

Leg length of the third square = 0.5·√2 ÷ 2 = 0.25·√2

Side length of the third square = √((0.25·√2)² + (0.25·√2)²) = 0.5

Leg length of the fourth square = 0.5 ÷ 2 = 0.25

Side length of the fourth square = √(0.25² + 0.25²) = 0.25·√2

The sequence of the sides of the first 4 terms is therefore;

• 1,  0.5·√2,  0.5,  0.25·√2

(b) The sequence for the areas for the first 4 terms are;

1², (0.5·√2)², 0.5², (0.25·√2)², which gives;

• 1,  0.5,  0.25,  0.125

c. The given sequence of the areas is a geometric progression, G.P.

Where;

The first term, a = 1

The common ratio, r = 0.5

The sum of a G.P to infinity is given as follows;

$\displaystyle \sum \limits^{\infty}_{k = 0} a \cdot r^ k= \mathbf{ \dfrac{a}{1 - r}}$

Which gives;

$\displaystyle \sum \limits^{\infty}_{k = 0} 1 \times \frac{1}{2} ^ {k}= \dfrac{1}{1 - 0.5}= 2$

• The sum of the areas of the infinite squares = 2

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