If $p,\ q,\ r$ are in A.P., then show that $p^2( p+r),\ q^2( r+p),\ r^2( p+q)$ are also in A.P.

Given:  $p,\ q,\ r$ are in A.P.

To do: To show that $p^2( p+r),\ q^2( r+p),\ r^2( p+q)$ are also in A.P.

Solution:

$p,\ q,\ r$ are in A.P.

$\Rightarrow 2q=p+r$

$p^2( q+r)+r^2( p+q)$

$=p^2q+p^2r+r^2p+r^2q$

$=p^2q+r^2q+pr( p+r)$

$=p^2q+r^2q+pr( 2q)$

$=q( p^2+r^2+2pr)$

$=q( p+r)^2$

$=q( p+r)( p+r)$

$=q( p+r)2q$

$=2q^2( p+r)$

$=2q^2( r+p)$

$p^2( q+r)+r^2( p+q)=2q^2( r+p)$

$\Rightarrow  p^2( q+r),\ q^2( r+p),\ r^2( p+q)$ are in A.P.

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Updated on: 10-Oct-2022

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