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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Sequences and Series

NDA 2021 · Mathematics · Sequences and Series

Let p, q and 3 be respectively the first, third and fifth terms of...

Let p,qp, q and 3 be respectively the first, third and fifth terms of an A.P. Let dd be the common difference. If the product (pq)(p q) is minimum, then what is the value of dd ?

Options

  1. A1
  2. B38\frac{3}{8}
  3. C98\frac{9}{8}
  4. D94\frac{9}{4}

Correct answer

C. 98\frac{9}{8}

Step-by-step solution

Given that a=P,a3=P+2d=qa=P, a_3=P+2 d=q ....(i) a5=P+4d=3⇒P=3−4d∴q=3−4d+2d=3−d[ from (i) ] Now let S=Pq=(3−4d)(3−2d)S=8d2−18d+9⇒dsdd=16d−18=0⇒d=98⇒d2sdd2=16>0∴Pq is minimum when d=98. \begin{aligned} & a_5=P+4 d=3 \Rightarrow P=3-4 d \\ & \therefore q=3-4 d+2 d=3-d[\text{ from (i) }] \\ & \text{ Now let } S=P q=(3-4 d)(3-2 d) \\ & S=8 d^2-18 d+9 \\ & \Rightarrow \frac{d s}{d d}=16 d-18=0 \\ & \Rightarrow d=\frac{9}{8} \\ & \Rightarrow \frac{d^2 s}{d d^2}=16 > 0 \\ & \therefore P q \text{ is minimum when } d=\frac{9}{8} \text{. } \end{aligned}
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