The sum of the areas of the squares, if the pattern were ...

[leilsilver]

The sum of the areas of the squares, if the pattern were continued indefinitely, is a/b...wherea and b are both natural numbers and the fraction is as reduced as much as possible. How mubh is sqrt a-b? And the image is a graph where the line is y= -1/4x +1 and the first square is in quadrant one with and are of 1^2. Starting from the y-intersect point and the squares get smaller.

Can anyone help me with it?

 

[leilsilver]

I wonder if anyone was wondering how to figure this out...anyway...I figured it out myself, finally.

This is what it looks like. I found the sequence to where the boxes intersect with the line y=1/4x+1. It goes something like this, 1, 3/4, 9/16, 27/64,... which is just going up in the power of 3^n/4^n, only starting at zero. And then when I squared it, went up by 3^2n/4^2n. Only that was really irrelevant as it wanted the geometric infinite series. So I wrote the areas of the squares in addition form to find out how much it went up by. (I'm slow like that if you already got it figure out, I'm just telling you my thought process) It went: 1+ 9/16 + 81/256 +729/4096+ ....

And since it went up by 9/16:

S= 1/(1-9/16)

S=1/(7/16)

S= 16/7

so sqrt a-b is 16-7, which equals 9 which when square rooted equals 3. [/img]

 

[galactus]

\(\displaystyle \L\\\frac{\frac{3}{4}}{1-\frac{3}{4}}=3\)

 

[leilsilver]

True.