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The interior angles of a polygon are in A.P., the smallest angle is 75° and the common difference is 10°, then number of sides of the polygon are,

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4 or 18

None

answer is A.

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Detailed Solution

Given that,
The interior angles of a polygon are in A.P., the smallest angle is 75° and the common difference is 10°.
We know,
If a is the first term and d is common difference and n is

n t h

term of A.P. ,then sum of first n terms of AP is

s n = n 2 2 a + n − 1 d

.
Where,
The first term is a.
The common difference is d.
The number of terms are n.
We also know,
Sum of all interior angle of a polygon is

n − 2 1800

.
Therefore,

n − 2 180 = n 2 275 + n − 1 10 ⇒ 180 n − 360 = n 2 150 + 10 n − 10 ⇒ 180 n − 360 = n 2 140 + 10 n ⇒ 180 n − 360 = 70 n + 5 n 2

⇒ 0 = 70 n + 5 n 2 − 180 n + 360 ⇒ 0 = 5 n 2 − 110 n + 360 ⇒ 0 = n 2 − 22 n + 72

⇒ n 2 − 22 n + 72 = 0 ⇒ n 2 − 18 n − 4 n + 72 = 0 ⇒ n n − 18 − 4 n − 18 = 0 ⇒ n − 4 n − 18 = 0

\Rightarrow

n=4,18
Therefore, the polygon has 4 or 18 sides.
Hence, option 3 is correct.

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