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Question: If the\(p ^ { t h }\), \(q ^ { t h }\) and \(r ^ { t h }\)term of a G.P. are \(a , b , c\) respecti...

If thepthp ^ { t h }, qthq ^ { t h } and rthr ^ { t h }term of a G.P. are a,b,ca , b , c respectively, then aq−r⋅br−p⋅cp−qa ^ { q - r } \cdot b ^ { r - p } \cdot c ^ { p - q } is equal to.

A

0

B

1

C

abca b c

D
Answer

1

Explanation

Solution

Let ARp−1=aA R ^ { p - 1 } = a …..(i)

ARq−1=bA R ^ { q - 1 } = b …..(ii) and ARr−1=cA R ^ { r - 1 } = c …..(iii)

So aq−rbr−pcp−qa ^ { q - r } b ^ { r - p } c ^ { p - q } ={ARp−1}q−r{ARq−1}r−p{ARr−1}p−q= \left\{ A R ^ { p - 1 } \right\} ^ { q - r } \left\{ A R ^ { q - 1 } \right\} ^ { r - p } \left\{ A R ^ { r - 1 } \right\} ^ { p - q }

=A(q−r+r−p+p−q)R(pq−pr−q+r+qr−pq−r+p+pr−rq−p+q)= A ^ { ( q - r + r - p + p - q ) } R ^ { ( p q - p r - q + r + q r - p q - r + p + p r - r q - p + q ) }

=A0R0=1= A ^ { 0 } R ^ { 0 } = 1.

Note : Such type of questions containing terms of powers in cyclic order associated with negative sign, reduce to 1 mostly.