Equilateral and Equiangular Polygons

A polygon is a 2 dimensional geometric figure bound with straight sides. A polygon is called Equilateral if all of its sides are congruent. Common examples of equilateral polygons are a rhombus and regular polygons such as equilateral triangles and squares.  Now, a polygon is equiangular if all of its internal angles are congruent.  Some important facts to consider The only equiangular triangle is the equilateral triangle If P is an equilateral polygon that has more than three sides, it does not have to be equiangular. A rhombus with no right angle is an example of an equilateral but non-equiangular polygon.  Rectangles, including squares, are the only equiangular quadrilaterals Equiangular polygon theorem. Each angle of an equiangular n-gon is  $$\Bigg(\frac{n-2}{n}\Bigg)180^{\circ} = 180^{\circ} -   \frac{360^{\circ}}{n} $$ Viviani's theorem   Vincenzo Viviani (1622 – 1703) was a famous Italian mathematician. With his exceptional intelligence in math...
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Sequences and Series

 Let's take a look at some interesting sequences and series problems.

1. (USAMTS) Evaluate the value of 

$$S = \sqrt{1+\cfrac{1}{1^2}+\cfrac{1}{2^2}} + \sqrt{1+\cfrac{1}{2^2}+\cfrac{1}{3^2}} + ... + \sqrt{1+\cfrac{1}{1999^2}+\cfrac{1}{2000^2}}$$

Solution

Notice that $$1+\cfrac{1}{n^2}+\cfrac{1}{(n+1)^2} = \cfrac{n^4+2n^3+3n^2+2n+1}{n^2(n+1)^2} = \cfrac{n^2+n+1)^2}{n^2(n+1)^2} = \Bigg(1+\cfrac{1}{n(n+1)} \Bigg)^2$$

Thus, $$S= \Bigg(1+\cfrac{1}{1 \cdot 2} \Bigg) + \Bigg(1+\cfrac{1}{2 \cdot 3} \Bigg) + \Bigg(1+\cfrac{1}{3 \cdot 4} \Bigg) + ... + \Bigg(1+\cfrac{1}{1999 \cdot 2000} \Bigg)$$ $$= 1999 + \Bigg(1 - \cfrac{1}{2} \Bigg) + \Bigg(\cfrac{1}{2} - \cfrac{1}{3} \Bigg) + ... + \Bigg(\cfrac{1}{1999} - \cfrac{1}{2000} \Bigg) = 1999 + 1 - \cfrac{1}{2000} = \boxed{\cfrac{3,999,999}{2000}}$$


2. (HMMT) Find the value of 

$$S = \cfrac{1}{3^2+1} + \cfrac{1}{4^2+2} + \cfrac{1}{5^2+3} + ...$$

Solution

For this problem, we can use partial fraction decomposition and try to find an alternate form for the expression $\cfrac{1}{(n+2)^2+n}$.

$$\cfrac{1}{(n+2)^2+n} = \cfrac{1}{n^2+5n+4} = \cfrac{1}{3} \Bigg(\cfrac{1}{n+1} - \cfrac{1}{n+4} \Bigg)$$

Hence, we can calculate the infinite sum as

$$S = \cfrac{1}{3} \Bigg( \cfrac{1}{2} - \cfrac{1}{5} + \cfrac{1}{3} - \cfrac{1}{6} + \cfrac{1}{4} - \cfrac{1}{7} + \cfrac{1}{5} - \cfrac{1}{8} + ... \Bigg) = \cfrac{1}{3} \Bigg( \cfrac{1}{2} + \cfrac{1}{3} + \cfrac{1}{4} \Bigg) = \boxed{\cfrac{13}{36}}$$

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Equilateral and Equiangular Polygons

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