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  • If ap, aq, ax of an HP are a,b and c respectively then the value of  <img alt="(q-r)bc+(r-p)ac+(p-q)ab" src="/latex-image/?%28q-r%29bc&plus;%28r-

If ap, aq, ax of an HP are a,b and c respectively then the value of  (q-r)bc+(r-p)ac+(p-q)ab is

Option: 1

0


Option: 2

abc


Option: 3

a+b+c


Option: 4

None of these


Answers (1)

best_answer

Let the first term be a and common difference be d for corresponding AP.

So, a+(p-1)d=\frac{1}{a_p}

\Rightarrow a+(p-1)d=\frac{1}{a_p}                    ......(1)

Similarly,         a+(q-1)d=\frac{1}{b}           ......(2)

and              a+(r-1)d=\frac{1}{c}                    .....(3)

(1)-(2)

(p-q)d=\frac{1}{a}-\frac{1}{b}

\Rightarrow (p-q)d=\frac{b-a}{ab}

\Rightarrow (p-q)ab=\frac{b-a}{d}                ........(4)

Similarly,

(q-r)bc=\frac{(c-d)}{d}                    ......(5)

and

(r-p)ac=\frac{(a-c)}{d}                ......(6)

(4) + (5) + (6)

(p-q)ab+(q-r)bc+(r-p)ac =\frac{b-a+c-b+a-c}{d}=0

 

Posted by

manish painkra

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