Solveeit Logo
Login

Question

Question: If the \(p ^ { t h } , q ^ { t h }\) and \(r ^ { t h }\) term of an arithmetic sequence are a , b ...

If the pth,qthp ^ { t h } , q ^ { t h } and rthr ^ { t h } term of an arithmetic sequence are a , b and cc respectively, then the value of [a(q−r)[ a ( q - r ) + b(r−p)b ( r - p ) +c(p−q)]=+ c ( p - q ) ] =

A

1

B

−1- 1

C

0

D

½

Answer

0

Explanation

Solution

Suppose that first term and common difference of A.P.'s are and D respectively.

Now, pthp ^ { t h } term =A+(p−1)D=a= A + ( p - 1 ) D = a …..(i)

qthq ^ { t h } term =A+(q−1)D=b= A + ( q - 1 ) D = b ......(ii)

and rthr ^ { t h } term =A+(r−1)D=c= A + ( r - 1 ) D = c …..(iii)

So, a(q−r)+b(r−p)+c(p−q)a ( q - r ) + b ( r - p ) + c ( p - q )

=a{b−cD}+b{c−aD}+c{a−bD}= a \left\{ \frac { b - c } { D } \right\} + b \left\{ \frac { c - a } { D } \right\} + c \left\{ \frac { a - b } { D } \right\} =1D(ab−ac+bc−ab+ca−bc)=0= \frac { 1 } { D } ( a b - a c + b c - a b + c a - b c ) = 0.