Area A sequence of smaller squares is formed by connecting the midpoints of the sides of a larger square.

(a) If the area of the largest square is 1 square unit,
determine the first five terms of a sequence that
describes the area of each successive square.
(b) Use a formula to sum the areas of the first 10
squares.

I hope you made a sketch

side of square 1 = 1
area = 1

side of square 2 = √2/2
area = 1/2

side of square 3 = 1/2
area of square 3 = 1/4

side of square 4 = √2/4
area of square 4 = 2/16 = 1/8

areas are : 1 , 1/2, 1/4, 1/8 , 1/16 ...

looks like a GS , where a=1, r = 1/2

sum(10) = a(1 - r^10) / (1-r)
= 1(1 - (1/2)^10 )/(1/2)
= 2(1 - 1/1024)
= 2(1023/1024)
= 1023/512

please explain where th square root of 2 came from

(a) In this problem, we can observe that each successive square is formed by connecting the midpoints of the sides of the previous square. Let's denote the area of the largest square as A₀ and the areas of the successive squares as A₁, A₂, A₃, A₄, A₅.

The area of each successive square is half the area of the previous square because we are connecting the midpoints of the sides.

So, we have the following sequence of areas:
A₁ = 1/2 * A₀ = 1/2 * 1 = 1/2 square units
A₂ = 1/2 * A₁ = 1/2 * 1/2 = 1/4 square units
A₃ = 1/2 * A₂ = 1/2 * 1/4 = 1/8 square units
A₄ = 1/2 * A₃ = 1/2 * 1/8 = 1/16 square units
A₅ = 1/2 * A₄ = 1/2 * 1/16 = 1/32 square units

Therefore, the first five terms of the sequence that describes the area of each successive square are 1/2, 1/4, 1/8, 1/16, and 1/32 square units.

(b) To find the sum of the areas of the first 10 squares, we can use the formula for the sum of a geometric series:

Sum = a * (1 - r^n) / (1 - r)

In this case, the first term a is 1/2, the common ratio r is 1/2 as well, and the number of terms n is 10.

Plugging these values into the formula, we get:

Sum = (1/2) * (1 - (1/2)^10) / (1 - 1/2)
= (1/2) * (1 - (1/1024)) / (1/2)
= (1/2) * (1023/1024) / (1/2)
= (1023/1024) * 2
= 1023/512 square units

Therefore, the sum of the areas of the first 10 squares is 1023/512 square units.

To find the areas of the successive squares in this sequence, we can follow a step-by-step approach.

(a) The area of each square in this sequence can be determined by taking the square of the length of its sides. Let's denote the side length of the first square as 's1', which is the side length of the largest square. Since the largest square has an area of 1 square unit, its side length must be 1 unit.

The sequence then continues by forming smaller squares using the midpoints of the sides of the larger square. Each smaller square will have half the side length of the preceding square.

So, the second square will have a side length of s2 = s1/2 = 1/2 units, and its area will be (s2)^2 = (1/2)^2 = 1/4 square units.

Following the same pattern, we can find the side length and area for the subsequent squares:
- The third square, s3 = s2/2 = (1/2)/2 = 1/4 units, with an area of (s3)^2 = (1/4)^2 = 1/16 square units.
- The fourth square, s4 = s3/2 = (1/4)/2 = 1/8 units, with an area of (s4)^2 = (1/8)^2 = 1/64 square units.
- The fifth square, s5 = s4/2 = (1/8)/2 = 1/16 units, with an area of (s5)^2 = (1/16)^2 = 1/256 square units.

Therefore, the first five terms of the sequence that describes the area of each successive square are: 1, 1/4, 1/16, 1/64, 1/256.

(b) To find the sum of the areas of the first 10 squares, we can use a formula for the sum of a geometric sequence. In this case, the common ratio between the terms (r) is 1/4 (since each term is obtained by dividing the previous term by 4).

The sum of a geometric sequence with the first term (a) and the common ratio (r), for a total of 'n' terms, can be calculated using the formula:

sum = a * (1 - r^n) / (1 - r)

Using this formula, we can substitute the values into the equation:

sum = 1 * (1 - (1/4)^10) / (1 - 1/4)

Evaluating this expression will give us the sum of the areas of the first 10 squares in the sequence.