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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Sequences and Series

NDA 2019 · Mathematics · Sequences and Series

A geometric progression (GP) consists of 200 terms. If the sum of...

A geometric progression (GP) consists of 200 terms. If the sum of odd terms of the GP is mm, and the sum of even terms of the GP is nn, then what is its common ratio?

Options

  1. Am/nm / n
  2. Bn/mn / m
  3. Cm+(n/m)m+(n / m)
  4. Dn+(m/n)n+(m / n)

Correct answer

B. n/mn / m

Step-by-step solution

∵Sn=a(1−rn)1−r\because S_n=\frac{a\left(1-r^n\right)}{1-r} given: a1+a3+…+a199=m∴a[(r2)100−1]r2−1=ma_1+a_3+\ldots+a_{199}=m\therefore \quad \frac{a\left[\left(r^2\right)^{100}-1\right]}{r^2-1}=m ...(1) also a2+a4+…+a200=n∴ar⁡[(r2)100−1]r2−1=n\begin{aligned} & a_2+a_4+\ldots+a_{200}=n \\ & \therefore \quad \frac{\operatorname{ar}\left[\left(r^2\right)^{100}-1\right]}{r^2-1}=n \end{aligned} ...(2)  eq. (2)÷(1)⇒r=nm\text{ eq. }(2) \div(1) \Rightarrow r=\frac{n}{m}
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