Question 1079865
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The angles of polygon are in arithmetic expression 172°,168° and 164°. How many sides does the polygon have?
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Your formulation is not perfect, unfortunately.


The correct formulation is this: 

<pre>
     The angles of polygon are in arithmetic progression 172°,168°, 164° and so on . . . . 
     How many sides does the polygon have?
</pre>

<U>Solution</U>


<pre>
The corresponding sequence of exterior angles is 8°, 12°, 16°  and so on . . . 


It is an arithmetic progression with the first term of 8 and the common difference of 4. 


The sum of exterior angles of any (convex) polygon is 360°.

So, you need to find "n", the number of sides/vertices, from the condition 

{{{S[n]}}} = 360°, where {{{S[n]}}} is the sum of the first n terms of this AP.


You can use the formula for {{{S[n]}}} = {{{(a[1]+((n-1)*d)/2)*n}}},

which gives you an equation

{{{(8 + ((n-1)*4)/2)*n}}} = 360,   or, which is the same

(8 + 2*(n-1))*n = 360.


It reduces to a quadratic equation

2n^2 + 6n - 360 = 0,   which is equivalent to

{{{n^2 + 3n - 180}}} = 0. 


It can be solved by factoring

(n-12)*(n+15) = 0,

which gives you only one positive solution n = 12.
</pre>

<U>Answer</U>.  n= 12.  The polygon has 12 sides.