a polygon has 31 sides,the lengths of which,starting from the smallest,are in AP.if the perimeter of the polygon is527cm and the length of the largest side is sixteen times the smallest,find the length of the smallest side and the common difference of the AP
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Answer:
Common difference of the AP is 1 and the smallest side of the polygon is 2 cm.
Step-by-step explanation:
A polygon has 31 sides are in Arithmetic progression.
Smallest side of the polygon is x cm
Since explicit formula of the arithmetic progression is
where a = first term = x
n = number of term
d = common difference
+ d
Perimeter of the polygon = x + (x + d) + (x + 2d)..........(x + 30d)
Sum of n terms of the arithmetic progression is represented by S(n) =
=
=
Since sum of all sides = perimeter of the polygon
31(x + 15d) = 527
x + 15d =
x + 15d = 17 -------(1)
Since largest side is 16 times of the smallest side.
x + 30d = 16x
16x - x = 30d
15x = 30d
x = 2d -------(2)
By placing the value of x from equation (2) to equation (1).
2d + 15d = 17
17d = 17
d = 1
From equation (2),
x = 2×1
x = 2 cm
Therefore, common difference of the AP is 1 and the smallest side of the polygon is 2 cm.
Learn more about Arithmetic Progression from brainly.in/question/7592978