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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Geometric Sequences

Let's take a look at geometric sequences today. In a way, they are similar to Arithmetic Sequences, but just think common ratio instead of common difference. Let's begin by taking a look at the geometric sequence $2, 6, 18, 54...$. The initial term, or the first term is $2$. The common ratio is defined as the number you multiply to each term to get the next one. In this case, the common ratio would be $3$, because $2 \cdot 3 = 6, 6 \cdot 3 = 18, 18 \cdot 3 = 54$, and so on. Common problems include finding the $n$th term, or finding the sum of such sequences.

To find the $n$th term of a geometric sequence, we use the formula:
$$a_n = ar^{n-1}$$
Where
$a_n =$ the $n$th term in the sequence
$a =$ the initial term
$r =$ the common ratio

Your Turn!
$1.$ Find the $7$th term in the sequence $5, 10, 20, 40...$
(Scroll down to the bottom of the page for the answer)

Next, let's take a look at how to find the sum of such a sequence. Let's find the sum of the sequence $$S_n = a + ar + ar^2 + ar^3 + ... + ar^{n-1}$$                                                   
Multiplying both sides by $r$, we get
$$rS_n = ar + ar^2 + ar^3 + ar^4 + ... ar^n$$                                                       
We can now subtracting the first equation from the second, noting that many terms cancel, we get
$$rS_n - S_n = (r-1)S_n = (ar + ar^2 + ar^3 + ar^4 + ... ar^n) - (a + ar + ar^2 + ar^3 + ... + ar^{n-1}) = ar^n-a$$
Dividing by $r-1$ to isolate $S_n$, we get the sum of a geometric sequence up to the $n$th term is
$$S_n = \frac{a(r^n-1)}{r-1}$$
Let's put this formula to use! To find the sum of the first $5$ terms of the sequence $2, 6, 18, 54...$, we know that the first term $ a = 2$, the common ratio $r = 3$, and $n=5$, because we want the sum of the first $5$ terms. Plugging these values into our formula, we get
$$S_n = \frac{a(r^n-1)}{r-1} \Rightarrow S_5 = \frac{2(3^ 5-1)}{3-1} = \frac{2 \cdot 242}{2} = \boxed{242}$$

Your Turn!
$2.$ Find the sum of the first $6$ terms of the sequence above.
(Scroll down to the bottom of the page for the answer)

The last thing that we will take a look at today is the sum of an infinite geometric sequence. If the absolute value of the common ratio is less than $1$, than we can have an infinite sequence. In other words, if $|r| < 1$, then we can find the sum of the geometric sequence. This makes sense because if the terms keep getting bigger, than the terms just approach infinity, and we can't find the sum. However, if the terms keep getting smaller, than it is possible to find the sum of all terms in the sequence, if it were to continue forever. The formula is
$$S_{\infty} = \frac{a}{1-r}$$

For example, if we had to find the infinite sum of a sequence with first term $6$, and the common ratio $\frac{1}{2}$, we would plug these values into our formula and get
$$\frac{a}{1-r} = \frac{6}{1-\frac{1}{2}} = \frac{6}{\frac{1}{2}} = \boxed{12}$$

Your Turn!
$3.$ Find the sum of the infinite sequence $2+\frac{2}{3} + \frac{2}{9}+\frac{2}{27}...$
(Scroll down to the bottom of the page for the answer)






Answers
$1. \text{ }320$
$2. \text{ }728$
$3. \text{ }3$

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Equilateral and Equiangular Polygons

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