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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Sequences and Series

NDA 2017 · Mathematics · Sequences and Series

If y=x+x^2+x^3+ up to infinite terms where x < 1 , then which one...

If y=x+x2+x3+…y=x+x^2+x^3+\ldots up to infinite terms where x<1\mathrm{x} < 1, then which one of the following is correct?

Options

  1. Ax=y1+y\mathrm{x}=\frac{\mathrm{y}}{1+\mathrm{y}}
  2. Bx=y1−yx=\frac{y}{1-y}
  3. Cx=1+yy\mathrm{x}=\frac{1+\mathrm{y}}{\mathrm{y}}
  4. Dx=1−yy\mathrm{x}=\frac{1-\mathrm{y}}{\mathrm{y}}

Correct answer

A. x=y1+y\mathrm{x}=\frac{\mathrm{y}}{1+\mathrm{y}}

Step-by-step solution

y=x+x2+x3+…y=x+x^2+x^3+\ldots upto infinite terms. x+x2+x3+…x+x^2+x^3+\ldots is a G.P. with a=xandr=xa=x and r=x. ∴y=x1−x⇒y−xy=x\begin{aligned} & \therefore y=\frac{x}{1-x} \\ & \Rightarrow y-x y=x\end{aligned}⇒x+xy=y⇒x(1+y)=y⇒x=y1+y\begin{aligned} & \Rightarrow x+x y=y \\ & \Rightarrow x(1+y)=y \\ & \Rightarrow x=\frac{y}{1+y}\end{aligned}$
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