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SEQUENCES AND SERIES — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSEQUENCES AND SERIES3 Marks
Question
The difference between any two consecutive interior angles of a polygon is $5^{\circ}$. If the smallest angle is $120^{\circ}$. find the number of the sides of the polygon.
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Answer

Let the number of sides of polygon be $n$. The interior angles of the polygon form an A.P.
Here, $a=120^{\circ}$ and $d=5^{\circ}$
Since, Sum of interior angles of a polygon with $n$ sides is $(n-2) \times 180^{\circ}$
$\therefore S_n=(n-2) \times 180^{\circ}$
$\Rightarrow \frac{n}{2}[2 \times 120+(n-1) \times 5]=180 n-360$
$\Rightarrow 120 n+\frac{5 n^2-5 n}{2}=180 n-360$
$\Rightarrow 240 n+5 n^2-5 n=360 n-720$
$\Rightarrow 5 n^2-125 n+720=0$
divide by 5 , we get
$\Rightarrow n^2-25 n+144=0$
$\Rightarrow(n-16)(n-9)=0$
$\Rightarrow n=16 \text { or } n=9$
But $\mathrm{n}=16$ not possible because $\mathrm{a}_{16}=\mathrm{a}+15 \mathrm{~d}=120+15 \times 5=195^{\circ}>180^{\circ}$
Therefore, number of sides of the polygon are 9

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