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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Sequences and Series

NDA 2023 · Mathematics · Sequences and Series

If (a+b), 2 b,(b+c) are in HP, then which one of the following is...

If (a+b),2b,(b+c)(a+b), 2 b,(b+c) are in HP, then which one of the following is correct?

Options

  1. Aa,b\mathrm{a}, \mathrm{b} and c are in AP
  2. Ba−b,b−c\mathrm{a}-\mathrm{b}, \mathrm{b}-\mathrm{c} and c−a\mathrm{c}-\mathrm{a} are in AP
  3. Ca,b\mathrm{a}, \mathrm{b} and c are in GP
  4. Da−b,b−c\mathrm{a}-\mathrm{b}, \mathrm{b}-\mathrm{c} and c−a\mathrm{c}-\mathrm{a} are in GP

Correct answer

C. a,b\mathrm{a}, \mathrm{b} and c are in GP

Step-by-step solution

since, a+b,2b,b+ca+b, 2 b, b+c are in H.P ⇒1a+b,12b,1b+c\Rightarrow \frac{1}{a+b}, \frac{1}{2 b}, \frac{1}{b+c} are in A.P ⇒2a+b,1b,2b+c\Rightarrow \frac{2}{a+b}, \frac{1}{b}, \frac{2}{b+c} are in A.P ⇒1b−2a+b=2b+c−1b⇒1b=a+2b+c(a+b)(b+c)⇒ab+ac+b2+\Rightarrow \frac{1}{b}-\frac{2}{a+b}=\frac{2}{b+c}-\frac{1}{b} \Rightarrow \frac{1}{b}=\frac{a+2 b+c}{(a+b)(b+c)}\Rightarrow \mathrm{ab}+\mathrm{ac}+\mathrm{b}^2+\mathrm{bc}=\mathrm{ab}+2b2+bc⇒b2=ac⇒a,b,c+2 \mathrm{ b}^2+\mathrm{bc}\Rightarrow \mathrm{b}^2=\mathrm{ac}\Rightarrow \mathrm{a}, \mathrm{b}, \mathrm{c} are in G.P.
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