<p>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 of the sides of S2. Assume the process continues indefinitely, with the vertices of Sk+1 being the midpoints of the sides of S1 for every positive integer k. What is the sum of the areas, in square inches, of S, S2, S3,…? </p>
<p>F. 40/3
G. 20
H. 70
J. 400/3
K. 200</p>
<p>This looks like a summation series (Σ). </p>
<p>The permitter is 40 so each side is 10 so area S1 = 100. This rules out F,G,H</p>
<p>If i’m understanding it correctly the corners of each new square are on the midpoint of the old one. The sides of S1 are 10, so the first new square’s side lengths can be determined using pythag as it forms a right triangle with short sides 5 & 5. This means that the side is √(5^2 + 5^2) =√50 and the area s2 = (√50)^2 = 50 vs s1 = 100. This first step leads us to think that area Sk+1 = Sk/2. Check with S3 where (√((√50/2)^2+(√50/2)^2))^2 =25 !!!</p>
<p>so we can use Σ100(1/2)^n = 200. To do this without an advanced calculator just estimate the first 5ish terms which are 100+50+25+13+6 =194.</p>
<p>Was this really a question on there? It seems to hard. It took me more than a minute to do it! I would have just eliminated the F,G,H using the first part and guessed J or K and then only came back at the end of the section if I had time.</p>
<p>yes, it was #60 on a released test. and the answer was 200. I looked it up and I guess there is a formula for a infinite series that would make it really easy if you knew it * (which i didn’T). </p>
<p>Thanks for your help!</p>
<p>Here’s a short cut to the fact that each square is twice as big as the next smaller one.</p>
<p>Connect the vertices of the smaller square; it comprises four congruent right triangles.
The large square is made up of eight such triangles. That’s all.
The area of the square S1 is 100, S2 - 50, S3 - 25, S4 - 12.5, S5 - 6.25 …
Student2407 made a very good point: on the ACT it’s often enough to evaluate the answer to figure out what the correct choice is - you could do this question without the formula.
194 is closer to 200 than to 400/3, thus the answer is K.</p>
<p>The perimeter is 40 so each side of the square is 10. Then, using the pythagorean theorm, you can find the sides of the second square.
You therefore have the following list for areas:
100, 50, …
the formula for an infinite geometric series is : Sum= Term1/ (1-ratio)
The Ratio is 50/100 (.5) and the first term is 100. Therefore the answer is k 200</p>
<p>@1eye651 - it’s a good practice to read prior posts before submitting your own. student2404 has already offered this solution.</p>