Permutations and Combinations

Permutations and Combinations

Question

The interior angles of a polygon are in A.P. The smallest angle is 120° and the common difference is 5°. Find the number of sides of the polygon.

Answer

Let the number of sides of the polygon be n. The interior angles from an A.P. Let a be the smallest angle and d be the common difference.
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Since the sum of all the interior angles of a polygon with n sides is left parenthesis straight n minus 2 right parenthesis space 180 degree
∴            straight S subscript straight n space equals space left parenthesis straight n minus 2 right parenthesis cross times 180
rightwards double arrow        space space space straight n over 2 left square bracket 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d right square bracket space equals space 180 straight n minus 360      rightwards double arrow   straight n over 2 left square bracket 2 left parenthesis 120 right parenthesis space plus space left parenthesis straight n minus 1 right parenthesis 5 right square bracket space equals space 180 straight n space minus space 360
rightwards double arrow       120 straight n space plus space fraction numerator 5 straight n squared minus 5 straight n over denominator 2 end fraction space equals space 180 straight n space minus space 360     rightwards double arrow  space space space 240 straight n space plus space 5 straight n squared minus 5 straight n space equals space 360 straight n space minus space 720
rightwards double arrow   5 straight n squared minus 125 straight n plus 720 space equals space 0                           rightwards double arrow       straight n squared minus 25 straight n plus 144 equals 0
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rightwards double arrow        space space left parenthesis straight n minus 9 right parenthesis space left parenthesis straight n minus 16 right parenthesis space equals space 0                          rightwards double arrow                           n = 9,  16
But if n = 16, then straight T subscript 16 space equals space straight a space plus space 15 straight d space equals space 120 degree space plus space 15 left parenthesis 5 degree right parenthesis space equals space 195 degree comma which is impossible.

∵  No angles of polygon exceeds 180 degree.
Hence,                         n = 9

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