First slide

Arithmetic progression

Question

The interior angles of a convex polygon are in A.P. If the smallest angle is $100 °$ and the common difference is $4 °$ , then the number of sides is

Moderate

A

5
B

7
C

36
D

44

Solution

We know that the sum of the interior angles of a convex polygon of $n$ sides is $(n - 2) (180 °)$ . We are given $a = 100$ and $d = 4 .$
$\begin{array}{l} ∴ \frac{n}{2} [2 (100) + (n - 1) (4)] = (n - 2) (180) \\ \Rightarrow n^{2} - 41 n + 180 = 0 \Rightarrow n = 5, 36 \end{array}$
If $n = 36,$ then interior angles of the polygon are $100 °, 104 °, . . . . . . . . ., 176 °, 180 °, 184 °, . . . . . . 244 ° .$

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