Class 11

Math

Algebra

Sequences and Series

If $a1 ,b1 ,c1 $ are the $p_{th},q_{th},r_{th}$ terms respectively of an $A.P$ the the value of $ab(p−q)+bc(q−r)+ca(r−p)$ is

- $−1$
- $2$
- $0$
- $−2$

We know foe any $n_{th}$ term of an $A.P$, $n_{th}$ term $=a+(n−1)d$ where $a=$ first term and $d=$ common difference.

$∴a_{1}+(p−1)d=a1 $,

$a_{1}+(9−1)d=h1 $ and $a_{1}+(r−1)d=c1 $

$∴a1 =(a_{1}−d)+pd−−−(1),b1 =(a_{1}−d)+qd−−−(2)$ and $c1 =(a_{1}−d)+rd−−−(3)$

By $(1)−(2),(2)−(3)$ and $(3)−(1)$

$a1 −b1 =(p−q)d,b1 ,c1 =(q−r)d$ and

$c1 −a1 =(r−p)d$

$∴d(ab)b−a =p−q,d(bc)c−b =q−r,d(ac)a−c =r−p$

or, $db−a =ab(p−q),dc−b =bc(q−r),da−c =ac(r−p)$

Adding all the equation

$d(b−a)+(c−b)+(a−c) =ab(p−q)+bc(q−r)+ac(r−p)$

or, $ab(p−q)+bc(q−r)+ac(r−p)=0$------Proved,