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KVPY2017MathematicsSequences and Series

Let n 4 be a positive integer and let ₁, ₂, . ., _ n be the lengths of the sides of arbitrary n - sided non-degenerate polygon P. Suppose ₁ ₂ + ₂ ₃ + _ n -1 _ n + _ n ₁ = n Consider the following statements: I. The lengths of the sides of P are equal. II. The angles of P are equal. III. P is a regular polygon if it is cyclic. Then

Options

  1. AI is true and I implies II
  2. BI is true
  3. CII is false
  4. DI and III are true

Correct answer

D. I and III are true

Step-by-step solution

given : ₁ ₂ + ₂ ₃ .+ _ n ₁ = n .........(i) Use A.M G.M We get ( ₁ ₂ + ₂ ₃ + _ n ₁ ) n [n] ₁ ₂ ₂ ₃ _ n ₁ n n 1 n = n So A M = G M Hence ₁ ₂ = ₂ ₃ .= _ n ₁ = k array l k = ₁+ ₂ + _ n ₂+ ₃ + _ n + ₁ =1 ₁= ₂ = _ n array

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