 Hello and welcome to the session. Let us discuss the following problem which says the difference between any two consecutive interior angles of a polygon is 5 degrees. If the smallest angle is of 120 degrees, find the number of sides of the polygon. So, let us begin with the solution. Since the smallest angle of the polygon is of 120 degrees and the difference between any two consecutive interior angles is 5 degrees, therefore the angles of the polygon forms an AP. The AP series is 120 degrees, then we have 125 degrees, then 130 degrees, then we have 135 degrees and so on. We have to find the number of sides polygon. Now the sum of n interior angles given by n upon 2 into 2a plus n minus 1 into d, where a is the first term of the series, d is the common difference which is equal to 5 cents. The difference between any two consecutive interior angles is 5 degrees. Now the sum of n interior angles of a polygon is given by 180 degrees into n minus 360 and this is equal to n upon 2 into 2a plus n minus 1 into d, where n is the number of angles of the polygon. And since n of polygon, the number of sides is equal to the number of angles, therefore we shall be finding this n and that will be equal to the number of sides. Now let us substitute the values of a and d. So we have 180 degrees into n minus 360 is equal to n upon 2 into 2 into 120 plus n minus 1 into d is 5. Or we further have 180 n minus 360 is equal to n upon 2 into 240 plus 5n minus 5 or we further have 180 n minus 360 is equal to n upon 2 into 240 minus 5 is 235 plus 5n or this can further be written as taking 5 common from the LHS we have 36n minus 72 is equal to not taking 5 common from there, we have 5n upon 2 into inside the placket 47 plus n. Now 5 cancels out with 5 or we have 36n minus 72 multiplied by 2 on cross multiplying is equal to n into 47 plus n or we have 36n into 2 minus 72 into 2 is equal to 47n plus n square or this can further be written as on simplifying n square plus 47n minus 72n plus 144 is equal to 0 or we have n square minus 25n plus 144 is equal to 0 or now splitting the middle term can be written as n square minus 16n minus 9n plus 144 is equal to 0 or now taking n common from the first two terms we have n minus 16 minus 9 into n minus 16 is equal to 0 or we have n minus 16 into n minus 9 is equal to 0 so this implies either n is equal to 16 or 9 now reject n is equal to 16 now let's see why we have rejected n is equal to 16 let us possible n is equal to 16 then the AP is 120 125 130 135 and so on up to 190 and 195 just try to make the polygon with these angles the first angle is of 120 degrees then the next angle is of 125 degrees the next angle is 130 degrees and since the angles are going on increasing so a stage will come and the two sides form a straight line suppose this is 180 degrees and after that we have 185 degrees so this angle will look like this and then we have 190 degrees and then we have 195 degrees so this is a figure which is formed by this angle now as we can see this figure is not close since the n points are not intersecting each other therefore n is equal to 16 is not possible you try to make it with the help of a scale and a d so it will become more clear to you hence our answer is only n is equal to 9 that is the number of sides of the polygon is equal to 9 so this completes the session hope you have understood it take care and have a good day