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Manipal MET2020MathematicsSequences and Series

If a, b and c are in HP, then for any n N , which one of the following is true?

Options

  1. Aa^n+c^n 2 b^n
  2. Ba^n+c^n 2 b^n
  3. Ca^n+c^n=2 b^n
  4. DNone of these

Correct answer

B. a^n+c^n 2 b^n

Step-by-step solution

Given that, a, b and c are in HP. Harmonic mean of a and c is b . and geometric mean of a and c is a c . Geometric mean Harmonic mean a c b ....(i) Now, for the positive numbers a^n and c^n , we have Geometric mean = a^n c^n and arithmetic mean = a^n+c^n 2 AM GM array ll & a^n+c^n 2 a^n c^n & a^n+c^n 2 a^n c^n & a^n+c^n 2( a c )^n 2 b^n [using Eq. (i)] & a^n+c^n 2 b^n array

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