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NDA 2023 Maths · Mathematics · Sequences and Series · easy

If a,b,c are in AP, then 2^a,2^b,2^c are in GP. 2. The sum of the first n odd natural numbers is n^2.

Single correct option

Consider the following statements:
1. If a,b,ca,b,c are in AP, then 2a,2b,2c2^a,2^b,2^c are in GP.
2. The sum of the first nn odd natural numbers is n2n^2.
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: (C) Both 1 and 2

If b−a=c−bb-a=c-b then 2b/2a=2b−a=2c−b=2c/2b2^b/2^a=2^{b-a}=2^{c-b}=2^c/2^b, so the three powers have a common ratio: statement 1 is correct. For statement 2, 1+3+⋯+(2n−1)=n2 (1+2n−1)=n21+3+\cdots+(2n-1)=\dfrac n2\,(1+2n-1)=n^2, which is also correct. Both hold, so the answer is both 1 and 2.
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Original question written in the NDA 2023 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