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NDA 2019 Maths · Mathematics · Sequences and Series · medium

For two distinct positive numbers, the arithmetic mean is less than the harmonic mean. 2. The sum to infinity of the GP 1-1/2+1/4-… is 2/3.

Single correct option

Consider the following statements:
1. For two distinct positive numbers, the arithmetic mean is less than the harmonic mean.
2. The sum to infinity of the GP 1−12+14−⋯1-\dfrac12+\dfrac14-\cdots is 23\dfrac23.
Which of the statements given above is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
Show answer and solution

Answer: (B) 2 only

For positive numbers AM≥GM≥HMAM\ge GM\ge HM, with strict inequality for distinct numbers, so the arithmetic mean is greater than the harmonic mean; statement 1 is false. Statement 2: a=1a=1 and r=−12r=-\dfrac12 with ∣r∣<1|r|<1, so S∞=11+12=23S_\infty=\dfrac{1}{1+\frac12}=\dfrac23; it is true. Hence only statement 2 is correct.
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Original question written in the NDA 2019 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake