Provide an appropriate response.The series of sketches below starts with a square having sides of length 1 (one). In the following steps, squares are constructed by joining, in order, the midpoints of the sides of the previous square.Show that the area of the nth new square is n, for all natural numbers n.

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible answer is below.

First we show that the given statement is true for n = 1. Using the Pythagorean Theorem, we see that each side of the first new square (n = 1) has a length of , so the area of the first new square is  ² =   =  = ¹. 

Next, we assume that for some unspecified natural number n, P_k: area of nth new square is 
Notice the four isosceles triangles contained in the nth new square but not contained in the (n+1)th new square. Because of the construction from midpoints, each of these triangles has an area equal to  of the area of the nth new square (the square containing them). Since we know the area of the nth new square to be  ⁿ, the area of the (n+1)th new square is equal to the area of the larger square minus the area of the four triangles, or  ⁿ- 4 ⁿ. After factoring this becomes
  or  ⁿ⁺¹