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BITSAT 2011
Exam
BITSAT
Year
2011
Subject
Mathematics
Chapter
Sequences and Series

BITSAT 2011 · Mathematics · Sequences and Series

If ( ^ th , ^ th ) and ( ^ th ) terms of H.P. are ( u, v, w...

If (\mathrmp^{\text{th }}, \mathrmq^{\text{th }}) and (\mathrmr^{\text{th }}) terms of H.P. are (u,v,w\mathrm{u}, \mathrm{v}, \mathrm{w} respectively, then find the value of the expression (q−r)vw+(r−p)wu+(p−q)uv(q-r) v w+(r-p) w u+(p-q) u v.

Options

  1. A2
  2. B0
  3. C4
  4. D8

Correct answer

B. 0

Step-by-step solution

Let H.P. be (\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{a}+\mathrm{d}}+\frac{1}{\mathrm{a}+2 \mathrm{ d}}+\ldots \ldots) $\begin{aligned}& \therefore \mathrm{u}=\frac{1}{\mathrm{a}+(\mathrm{p}-1) \mathrm{d}}, \mathrm{v}=\frac{1}{\mathrm{a}+(\mathrm{q}-1) \mathrm{d}},\\ & \mathrm{w}=\frac{1}{\mathrm{a}+(\mathrm{r}-1) \mathrm{d}} \Rightarrow \mathrm{a}+(\mathrm{p}-1) \mathrm{d}=\frac{1}{\mathrm{u}}\\ & \mathrm{a}+(\mathrm{q}-1) \mathrm{d}=\frac{1}{\mathrm{v}}, \mathrm{a}+(\mathrm{r}-1) \mathrm{d}=\frac{1}{\mathrm{w}}\\ & \Rightarrow(\mathrm{q}-\mathrm{r})\{\mathrm{a}+(\mathrm{p}-1) \mathrm{d}\}+(\mathrm{r}-\mathrm{p})\{\mathrm{a}+(\mathrm{q}-1) \mathrm{d}\}+\ldots \ldots \\ & =\frac{1}{\mathrm{u}}(\mathrm{q}-\mathrm{r})+\frac{1}{\mathrm{v}}(\mathrm{r}-\mathrm{p})+\ldots . \Rightarrow(q-r) \mathrm{vw}+\ldots . . =0 \end{aligned}$)
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