Problem:
https://www.desmos.com/calculator/hv0sfm020w
The problem presented in the provided link displays a rectangle that can be adjusted from the stages of $n=3$ up to $n=13$. With each increase in n, the length and width of the rectangle also increase. I have made an effort to formulate a mathematical identity and attempted to prove it, although I believe I may be incorrect.
Process
Each stage of "n" obviously has a length and width, and also has a few inner squares, each of which has a square root attached to it. Let's take the example of n=3. The length multiplied by width is 6, and the inner squares also have a sum of 6. To be precise, 2 to the power of 2 means 2 multiplied by 2, which equals 4. Similarly, 1 to the power of 2 is 1 multiplied by 1, which equals 1, and in n=3, there is three inner rectangles with each number having a square root, and those numbers are: 2 to the power of 2, 1 to the power of 2, and again 1 to the power of 2. Therefore, the sum of inner squares of n=3 is 4 + 1 + 1 = 6.This continues throughout each value of n, in another example, n= 8 would be length x width = 714, and the sum of 34 to the power of 2, 21 to the power of 2, 13 to the power of 2, 8 to the power of 2, 5 to the power of 2, 1 to the power of 2, and finally another 1 to the power of 2 would all add up to 714. These scenarios are not a coincidence; it is true for all values of n ranging from 3 to 13. As a result, my identity is:
Identity
$P(n)= A = B$ is true for all $n$
In the case that $A$ is length x width, and $B$ is the sum of the rectangle
It's clear that the identity holds for $n=3$, where 6 = 6, and I'm confident that it's true. However, extending it to all values of n seems daunting, especially for $n=13$, where the result is 84,071 = 84,071. I am struggling to understand how this identity can hold true for all values of $n$.
My thinking..Possible Proof
From my understanding, I am confident that my identity is valid for all values of n ranging from 3 up to 13. Therefore, my identity has been proven to be true. This implies that the equality P(n) = A = B holds true for all values of $n$ up to 13, where $P(13) = 84,071 = 84,071$.
Question
This leads to the question of how to find an identity of $n=3$, $n=4$, $n=5$ all the way up until $n=13$ and prove it....is there anything wrong with my work?