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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Sequences and Series

NDA 2016 · Mathematics · Sequences and Series

Paragraph: The interior angles of a polygon of n sides are in AP....

Paragraph: The interior angles of a polygon of n sides are in AP. The smallest angle is 120∘120^{{}^{\circ}} and the common difference is 5∘5^{{}^{\circ}}.
Question: How many possible values can n have?

Options

  1. AOne
  2. BTwo
  3. CThree
  4. DInfinitely many

Correct answer

A. One

Step-by-step solution

Here, a=120∘a=120^{{}^{\circ}} and d=5. Sum of angles of polygon =(n−2)180∘=(n-2) 180^{{}^{\circ}}⇒n2[2a+(n−1)d]=(n−2)180⇒n2[2×120+(n−1)5]=(n−2)180⇒n2−25n+144=0⇒(n−9)(n−16)=0∴n=9,16\begin{aligned} & \Rightarrow \frac{n}{2}[2 a+(n-1) d]=(n-2) 180 \\ & \Rightarrow \frac{n}{2}[2 \times 120+(n-1) 5]=(n-2) 180 \\ & \Rightarrow n^2-25 n+144=0 \\ & \Rightarrow(n-9)(n-16)=0 \\ & \therefore n=9,16 \end{aligned}$ForForn=9, T_9=120+(9-1)^{5}=160ForForn=16, T_{16}=120+(16-1)^{5}=195$ [not possible]
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