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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Progressions

Let us observe the following pattern of numbers.

     $(i)$   $5, 11, 17, 23, ............$
   $(ii)$   $6, 12, 24, 48, ............$
  $(iii)$   $4, 2, 0, -2,-4, ............$
  $(iv)$   $\cfrac {2}{3}, \cfrac{4}{9}, \cfrac{8}{27}, \cfrac {16}{81},......... $

In example, $(i)$, every number (except 5) is formed by adding $6$ to the previous numbers. Hence a specific pattern is followed in the arrangement of these numbers. Similarly, in example $(ii)$, every number is obtained by multiplying the previous number by $2$. Similar cases are followed in examples $(iii)$ and $(iv)$. 

SEQUENCE

 A systematic arrangement of numbers according to a given rule is called a sequence. 
   The numbers in a sequence are called its terms. We refer the first term of a sequence as $T_{1}$, the second term as $T_{2}$ and so on. The $n$th terms of a sequence is denoted by $T_{n}$, which may also be referred to as the general term of the sequence. 

 Finite and Infinite Sequences

 $1.$ A sequence which consists of a finite number of terms is called a finite sequence. 

        Ex:
       $(a)$  $3, 6, 9, 12, 15, 18, 21, 24$ is the finite sequence of 8 terms. 

 $2.$ A sequence which consists of a infinite number of terms is called an infinite sequence. 

        Ex:
       $(a)$   $2, 9, 16, 23, 30,.........$ is an infinite sequence.

 Note: If a sequence is given, then we can find its $nth$ term and if the $nth$ term of a sequence is given we can find the terms of the sequence. 

        Ex: 
       Find the first four terms of the sequences whose $nth$ terms are given as follows. 
       $(i)$   $T_{n} = 3n + 1$
                 Substituting $ n = 1$, 
                 $T_{1} = 3(1) + 1 = 4$

                  Similarly, $T_{2} = 3(2) + 1 = 7$
                                $T_{3} = 3(3) + 1 = 10$
                                $T_{4} = 3(4) + 1 = 13$

        Thus, the first four terms of the sequence are $4, 7, 10, 13$. 

       $(ii)$   $T_{n} = 2n^2 - 3$
                 Substituting $ n = 1$, 
                 $T_{1} = 2(1)^2 - 3 = -1$

                  Similarly,  $T_{2} = 2(2)^2 - 3 = 5$
                                 $T_{3} = 2(3)^2 - 3 = 15$ 
                                 $T_{4} = 2(4)^2 - 3 = 29$ 

        Thus, the first four terms of the sequence are $-1, 5, 15, 29$. 

        
 Sequences of numbers which follow specific patterns are called Progressions. Depending on the pattern the progressions are classified as follows each of which will be reviewed in separate posts. 
          
$(i)$ Arithmetic Progression
$(ii)$ Geometric Progression
$(iii)$ Harmonic Progression

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Equilateral and Equiangular Polygons

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