Progressions - 2 (IIFT-2008)

The interior angles of a polygon are in A.P. If the smallest angle is 120⁰ and common difference is 5⁰, then the number of sides in the polygon is:

(A) 7    (B) 8    (C) 9    (D)None of the above

Solution follows here:

Solution:

Given that smallest interior angle = 120⁰ and common difference 5⁰

∴the series of interior angles is 120⁰, 125⁰, 130⁰,….

Exterior angle = 180⁰ – Interior angle

∴the series of corresponding exterior angles is 60⁰,55⁰,50⁰,…

This is a decreasing Arithmetic Progression with

initial term ‘a’ = 60;           common difference ‘d’ = -5;

Let the polygon has ‘n’ sides.

Sum of n terms of AP  = n(2a+(n-1)d)/2                     ----- >(1)

Sum of exterior angles of any polygon = 360⁰        ----- >(2)

From (1) and (2),

360     = n(2a+(n-1)d)/2

                                                360     = n(120+(n-1)(-5))/2

                                                360     = n(125-5n)/2

                                                720     = 5n(25-n)

                                                144     = n(25-n)

Instead of solving the equation, we go for checking multiple choice options one by one:

Put n = 7, 144 = 7(18)      wrong

Put n = 8, 144 = 8(17)      wrong

Put n = 9, 144 = 9(16)      correct

∴ Answer is (C)