Question

a circle is completely divided into n sectors in such a way that the angles of the sectors are in arithtmetic progression. If the smallest of these angles is 18 degree and the largest 72 degree, calculate n and the angle in the fourth sector.

Answer 1

as the angles are in arithmetic progression, then we have

A = 18, An = 72, Sn = 360

An = A + (n-1)d

72 = 18 + (n-1)d

or (n-1)d = 54

d = 54/(n-1)..................................(1)

also we know that, Sn = n/2 [2a + (n-1)d]

360 * 2 = n [ 2 * 18 + (n-1) * (54/(n-1) ]---------------( using (1) )

720 = n [ 36 + 54 ]

n = 8.............. (2)

using (2) in (1) , we get :-

d = 54/7

now A4 = 18 + ( 8-1) * 54/7

A4 = 41.14 degrees