Question

If ${{\text{p}}^{\text{th}}}\text{,}{{\text{q}}^{\text{th}}}$ and ${{\text{r}}^{\text{th}}}$ term of an A.P are a, b and c respectively, then prove that $a(q-r)+b(r-p)+c(p-q)=0$ .

Answer 1

Hint: For solving this question, we will use the concept of ${{\text{n}}^{\text{th}}}$ term of an A.P whose first term is a and difference is d then ${{\text{n}}^{\text{th}}}$ term of an A.P is a + (n - 1)d . We will first evaluate the values of ${{\text{p}}^{\text{th}}}\text{,}{{\text{q}}^{\text{th}}}$ and ${{\text{r}}^{\text{th}}}$ term of an A.P and, then we will substitute the values of ${{\text{p}}^{\text{th}}}\text{,}{{\text{q}}^{\text{th}}}$ and ${{\text{r}}^{\text{th}}}$ term of an A.P in a given condition and check whether the condition is satisfied or not.

Complete step by step answer:
It is given that ${{\text{p}}^{\text{th}}}\text{,}{{\text{q}}^{\text{th}}}$ and ${{\text{r}}^{\text{th}}}$ term of an A.P are a, b and c respectively. So, if first term of an A.P is A and common difference is equals to d then,
\[a\text{ }=\text{ }A\text{ }+\text{ }\left( \text{ }p\text{ }\text{- }1\text{ } \right)\cdot d\]
Re – writing a as,
\[a=(A-d)+dp\]
\[\text{b }=\text{ }A\text{ }+\text{ }\left( \text{ q }\text{- }1\text{ } \right)\cdot d\]
Re – writing a as,
\[b=(A-d)+dq\]
\[\text{c }=\text{ }A\text{ }+\text{ }\left( \text{ r }\text{- }1\text{ } \right)\cdot d\]
Re – writing a as,
\[c=(A-d)+dr\]
Now, putting the value of a, b, c in condition $a(q-r)+b(r-p)+c(p-q)=0$, we get
[ ( A – d ) +dp ]{ q – r } + [ ( A – d ) + dq ]{ r – p } + [ ( A – d ) + dr ]{ p – q } = 0
On re – writing we get,
( A – d )( q – r ) + ( A – d )( r – p ) + ( A – d )( p – q ) + dp( q – r ) + dq( r – p ) + dr( p – q ) = 0
Taking common factor out of the equation,
 ( A – d ){ ( q – r ) + ( r – p ) + ( p – q ) } + d{ p( q – r ) + q( r – p ) + r( p – q ) } = 0
On simplifying, we get
( A – d ){ q – r + r – p + p – q } + d{ pq – pr + qr – qp + rp – rq } = 0
On cancelling opposite terms we get,
( A – d ){ 0 } + d{ 0 } = 0
Hence, L.H.S = R.H.S

Note: As we see on substituting the values of a, b and c, we get a very complex equation so try to avoid calculation error as it may affect the answer and change the value. Re – arranging of the terms should be done in such a way that calculation becomes a bit easy and terms get cancelled out.