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.
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To find the sequence that describes the area of each successive square, we can start by understanding the pattern formed by connecting the midpoints of the sides of the larger square.
Let's start with a larger square of side length S. When we connect the midpoints of each side of this square, we form four smaller squares with side lengths of S/2. The total area of these four smaller squares is equal to the area of the larger square.
Since we know that the area of the largest square is 1 square unit, we can create the sequence by dividing the area of each square by the total area of the largest square.
(a) Determine the first five terms of the sequence:
1. The area of the largest square is 1 square unit.
2. The total area of the four smaller squares is equal to the area of the largest square, which is 1 square unit. Therefore, each of the four smaller squares has an area of (1/4) square units.
3. The next larger square is made up of four even smaller squares, each with an area of (1/4)*(1/4) = (1/16) square units.
4. Continuing this pattern, the areas of the next two larger squares will be (1/16)*(1/16) = (1/256) square units and (1/256)*(1/256) = (1/65536) square units, respectively.
So, the first five terms of the sequence that describes the area of each successive square are 1, 1/4, 1/16, 1/256, 1/65536.
(b) To sum the areas of the first 10 squares, we can use a formula for the sum of a geometric series. The formula is:
Sn = a * (r^n - 1) / (r - 1)
Where Sn is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
In this case, the first term (a) is 1, the common ratio (r) is 1/4 (since each term is obtained by multiplying the previous term by 1/4), and we want to find the sum of the first 10 terms (n = 10).
Plugging in the values into the formula, we get:
S10 = 1 * (1/4^10 - 1) / (1/4 - 1)
= (1/1024 - 1) / (1/4 - 1)
= (1/1024 - 1) / (-3/4)
= (-1023/1024) / (-3/4)
= (1023/1024) * (4/-3)
= -1368/3072
= -57/128
Therefore, the sum of the areas of the first 10 squares is -57/128 square units.