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 #3
avatar+33653 
+4

I get the following

Jan 10, 2021
 #1
avatar+33653 
+3

As follows:

 

Jan 10, 2021
 #4
avatar+26388 
+1

If the given equilateral triangle has a side measure of 4.
What is the sum of the areas of the stacked squares inside it?

 

My attempt:

 

\(\text{Let}~ \tan(60^\circ) = \sqrt{3} \\ \text{Let the side of the equilateral triangle }~ s_0 = 4 \\ s_1~\text{side of stacked square 1} \\ s_2~\text{side of stacked square 2} \\ s_3~\text{side of stacked square 3} \\ s_4~\text{side of stacked square 4} \\ \ldots \\ s_n~\text{side of stacked square n} \\\)

\(\begin{array}{|rcll|} \hline s_0 &=& s_1 + 2x_1 \quad | \quad \tan(60^\circ) = \dfrac{s_1}{x_1}~ \text{or}~ s_1 = x_1\tan(60^\circ)=x_1\sqrt{3} \\ s_0 &=& x_1\sqrt{3} + 2x_1 \\ s_0 &=& x_1(2+\sqrt{3}) \\ x_1 &=& \dfrac{s_0}{(2+\sqrt{3})}\times \dfrac{(2-\sqrt{3})} {(2-\sqrt{3}) } \\ x_1 &=& \dfrac{s_0(2-\sqrt{3})} { (2+\sqrt{3})(2-\sqrt{3}) } \\ x_1 &=& \dfrac{s_0(2-\sqrt{3})} { 4-3 } \\ \mathbf{x_1} &=& \mathbf{s_0(2-\sqrt{3})} \\ \hline s_0 &=& s_1 + 2x_1 \quad | \quad \mathbf{x_1=s_0(2-\sqrt{3})} \\ s_0 &=& s_1 + 2s_0(2-\sqrt{3}) \\ s_1 &=& s_0 - 2s_0(2-\sqrt{3}) \\ s_1 &=& s_0\Big(1-2(2-\sqrt{3})\Big) \\ s_1 &=& s_0(1-4+2\sqrt{3}) \\ \mathbf{s_1} &=& \mathbf{s_0(2\sqrt{3}-3)} \\ \hline \end{array}\)


In the same way...

\(\begin{array}{|rcll|} \hline s_2 &=& s_1(2\sqrt{3}-3) \quad | \quad \mathbf{s_1=s_0(2\sqrt{3}-3)} \\ s_2 &=& s_0(2\sqrt{3}-3)(2\sqrt{3}-3) \\ s_2 &=& s_0(2\sqrt{3}-3)^2 \\ \hline \ldots \\ s_3 &=& s_0(2\sqrt{3}-3)^3 \\ s_4 &=& s_0(2\sqrt{3}-3)^4 \\ \ldots \\ \mathbf{s_n} &=& \mathbf{s_0(2\sqrt{3}-3)^n} \\ \hline \end{array} \)

 

sum

\(\begin{array}{|rcll|} \hline \mathbf{sum_4} &=& \mathbf{s_1^2+s_2^2+s_3^2+s_4^2} \\ sum_4 &=& s_0^2(2\sqrt{3}-3)^2+s_0^2(2\sqrt{3}-3)^4+s_0^2(2\sqrt{3}-3)^6+s_0^2(2\sqrt{3}-3)^8 \\ sum_4 &=& s_0^2\Big( \underbrace{ (2\sqrt{3}-3)^2+(2\sqrt{3}-3)^4+(2\sqrt{3}-3)^6+(2\sqrt{3}-3)^8\Big) }_{Geometric~Sequence~q=(2\sqrt{3}-3)^2} \\ sum_4 &=& s_0^2(q+q^2+q^3+q^4) \\ sum_4 &=& s_0^2\dfrac{q}{1-q} (1-q^4) \quad | \quad q = 0.2153903092 \\ sum_4 &=& 4^2\times 0.2745190528 \times 0.9978476909 \\ sum_4 &=& 4.3923048454 \times 0.9978476909 \\ \mathbf{sum_4} &=& \mathbf{4.3828512478} \\ \hline \end{array}\)

 

Infinite Geometric Sequence

\(\begin{array}{|rcll|} \hline sum_{\infty} &=& s_0^2\dfrac{q}{1-q} \quad | \quad q = 0.2153903092 \\ sum_{\infty} &=& 4^2\times 0.2745190528 \\ \mathbf{sum_{\infty}} &=& \mathbf{4.3923048454} \\ \hline \end{array}\)

 

laugh

Jan 10, 2021

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