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...
Post a Comment

Fibonacci Numbers

 The sequence of numbers that starts as $1,1,$ and continues with each new number calculated as the sum of the previous two numbers, i.e.,


$1,1,2,3,5,8,13,21,34,55,89,144....$

is called the sequence of Fibonacci numbers. It was first described in 1202 by Leonardo of Pisa, better known as Fibonacci. It appeared in his book Liber Abaci as the solution to a problem involving the growing number of rabbits over time. Since then, Fibonacci numbers have seen many interesting applications and connections to a variety of topics, both within mathematics and in other disciplines, such as computer science, physics, chemistry, biology, and architecture. 


This pattern of Fibonacci numbers is given by $u_1 = 1, u_2 = 1$ and the recursive formula

$u_n = u_{n-1} + u_{n-2}, n > 2$.


Simple Properties of Fibonacci Numbers


$1.$

Sum of the Fibonacci Numbers

The sum of the first n Fibonacci numbers can be expressed as

$u_1 + u_2 + ... + u_{n-1} + u_n = u_{n+2} − 1.$


$2.$

Sum of Odd Terms

The sum of the Odd Terms of a Fibonacci sequence can be expressed as

$u_1 + u_3 + u_5 +... + u_{2n-1} = u_{2n}.$


$3.$

Sum of Even Terms

The sum of the Even Terms of a Fibonacci sequence can be expressed as

$u_2 + u_4 + u_6 +... + u_{2n} = u_{2n+1} - 1.$


$4.$

Sum of Fibonacci Numbers with Alternating Signs

The sum of the Fibonacci numbers with alternating signs can be expressed as 

$u_1 - u_2 + u_3 - u_4 +...+(-1)^{n+1}u_n = (-1)^{n+1}u_{n-1} + 1.$


$5.$

Sum of Squares

The sum of the squares of the first n Fibonacci numbers can be expressed as 

$u^2_1 - u^2_2 +...+u^2_{n-1} + u^2_n = u_nu_{n+1}.$


$6.$

Difference of Squares of Fibonacci Numbers

$u_{2n} = u^2_{n+1} - u^2_{n-1} $

Comments

Post a Comment

Popular posts from this blog

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...
Read more

Irrational Numbers

A number which cannot be written in the form $ \cfrac {p}{q}$  where $p$ and $q$ are integers and $ q $  $ \neq 0$. Example: $ \sqrt {2}$, $ \sqrt {3}$, $ \sqrt {6}$, $ \sqrt {7}$, $ \sqrt {8}$, $ \sqrt {10}$ Theorem: If $p$ divides $x^3$, then $p$ divides $x$, where $x$ is a positive integer and $p$ is a prime number.  Proof :  Let $x = p_{1} p_{2}...p_{n}$ where $p_{1}, p_{2}, p_{3}.....p_{n}$  are primes, not necessarily distinct.  $ \rightarrow x^3 = p_{1}^3 p_{2}^3.....p_{n}^3$ Given that $p$ divides $x^3$ By fundamental theorem, $p$ is one of the primes of $x^3$.  By the uniqueness of fundamental theorem, the distinct primes of $x^3$ are same as the distinct primes of $x$.  $ \rightarrow p$ divides $x$ Similarly if $p$ divides $x^2$, then $p$ divides $x$, where $p$ is a prime number and $x$ is a positive integer. Example: Prove that  $ \sqrt {2}$ is irrational.  Solution : Let us assume that  $ \sqrt ...
Read more

Incenter/Excenter Lemma

Image
Given $\triangle ABC$ with incenter $I$, extend $AI$ to meet the circumcircle of $\triangle ABC$ at $L$. Then, reflect $I$ over $L$, and name this point $I_a$. Then:  $(i)$ $BICI_a$ is a cyclic quadrilateral with point $L$ as the center of the circumscribed circle.  $(ii)$ $BI_a$ and $CI_a$ bisect the exterior angles of $\angle B$ and $\angle C$,  respectively.  Proof $(i)$ Notice that the the question is equivalent to proving the distance from $L$ to points $B, I, C, I_a$ are equal. In other words, we need to show that  $$LB = LI = LC = LI_a$$ Firstly, it is obvious that $LI = LI_a$ because by definition, $I_a$ is the reflection of $I$ over $L$, and reflections preserve the length of the segment.  Now, we are left with proving that $LB=LI$, because similar calculations will provide $LI=LC$.  We see that most of our given information involves angles, so we focus on proving $\angle LBI = \angle LIB$. Firstly,  $$\angle LBC = \angl...
Read more