JEE Main 2025 Jan 28 Shift 2, Mathematics Q23: Arithmetic, Geometric And Harmonic Progressions

JEE Main2025Jan 28, Shift 2Mathematics

The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6°$ . If the largest interior angle of the polygon is $219°$ , then $n$ is equal to _____.

Your answer

Solution

Let the interior angles be

$a, a + 6, a + 12, \dots, 219$

Since the largest angle is $21 9^{\circ}$ ,

$a + 6 (n - 1) = 219$

$\Rightarrow a = 225 - 6 n$

Sum of interior angles of an $n$ -sided polygon is

$\frac{n}{2} (a + 219) = (n - 2) \times 180$

Substituting $a = 225 - 6 n$ ,

$\frac{n}{2} (444 - 6 n) = 180 (n - 2)$

$222 n - 3 n^{2} = 180 n - 360$

$3 n^{2} - 42 n - 360 = 0$

$n^{2} - 14 n - 120 = 0$

$(n - 20) (n + 6) = 0$

$∴ n = 20$

$20$

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JEE Main 2025 Jan 28 Shift 2, Mathematics Q23: Arithmetic, Geometric And Harmonic Progressions

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