# Thread: Summing areas of squares

1. ## Summing areas of squares

A square S1 has a perimeter of 40 inches. The vertices of a second square S2 are the midpoints of the sides of S1. The vertices of a third square S3 are the midpoints the sides of S2. Assume the process continues indefinitely, with the vertices of S K+1 being the midpoints of the sides of Sk for every positive integer k. What is the sum of the areas, in square inches, of S1, S2, S3,...?

My instructor says the answer is 200 but I still don't understand how???

2. Originally Posted by donnagirl
A square S1 has a perimeter of 40 inches. The vertices of a second square S2 are the midpoints of the sides of S1. The vertices of a third square S3 are the midpoints the sides of S2. Assume the process continues indefinitely, with the vertices of S K+1 being the midpoints of the sides of Sk for every positive integer k. What is the sum of the areas, in square inches, of S1, S2, S3,...?

My instructor says the answer is 200 but I still don't understand how???
Your perimeter is 40, so each side is 10

Then the first square has an area of 10*10 = 100

Now lets find the area of the second square. It will be equal to the area of the first square, minus the four triangles around the edges.

So Area = 100-4*(area of triangle)

We know the length and height, because they go to the midpoint, so the length will be 5, and the height will be 5.
So the area of the triangle is (1/2)bh = (1/2)5*5
Area = 100-4*(1/2)*5*5
Area = 100-50 = 50

Notice that this triangle's area is half the area of the first triangle. Continuing with this pattern, you will see that each triangle's area will be half of the area of the triangle it is bounded by.

so our total area = 100 + 50 + 25 + 12.5 + ...

= 100 + (1/2)100 + (1/2)(1/2)100 + (1/2)(1/2)(1/2)100 + ....

= (1/2)^0 *100 + (1/2)^1 *100 + (1/2)^2 *100 + (1/2)^2 *100 + ...

so we can write this as a geometric series (see Geometric series - Wikipedia, the free encyclopedia for more about the formulas)

$= \sum_{n=0}^\infty 100(1/2)^n$

Now in a geometric series, if |r| < 1, the series converges to $\frac a{1-r}$

In our case, r = 1/2, and a =100.
1/2 <1 so our series converges

$= \frac {100}{1-1/2}$

$= \frac {100}{1/2}$

$= 200 ~in^2$

3. Hello, donnagirl!

Another approach . . . same answer.

A square $S_1$ has a perimeter of 40 inches.
The vertices of a second square $S_2$ are the midpoints of the sides of $S_1.$
The vertices of a third square $S_3$ are the midpoints the sides of $S_2.$
Assume the process continues indefinitely, with the vertices of $S_{k+1}$
being the midpoints of the sides of $S_k$ for every positive integer $k.$
What is the sum of the areas, in square inches, of $S_1,\,S_2,\,S_3,\,\hdots$ ?
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The sides and diagonals of $S_{k+1}$ divides $S_k$ into eight congruent triangles.

Since $S_{k+1}$ is composed of four of these triangles,
. . the area of $S_{k+1}$ is one-half of $S_k$

That is, each square has half the area of the preceding square.

$S_1$ has a perimeter of 40 inches; its side is 10 inches.
. . Its area is: . $10^2 \,=\,100$ in²

Then the total area is: . $A \;=\;100 + \frac{100}{2} + \frac{100}{4} + \frac{100}{8} + \cdots$

. . $\text{And we have: }\;A \;=\;100\underbrace{\left(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots\right)}_{\text{geometric series}}$

The geometric series has the sum: . $S \:=\:\frac{1}{1-\frac{1}{2}} \:=\:2$

. . Therefore: . $T \;=\;100(2) \;=\;200\text{ in}^2$