Question

If a,b,c are in AP, show that
(i) (b+c) , (c+a) and (a +b) are in AP.
(ii) ` a^(2) (b+c) , b^(2) (c +a) and c^(2) ( a+b) are in AP.

Answer 1

Since , a,b,c are in AP. We have
2b = ( a+c) ,
if ( c+a) -(b +c) = ( a+b) -( c-a)
i.e, if a -b = b-c
i.e,2b = a+c which is true by (i),
a,b,c are in AP ` Rightarrow (b+c), ( c+a) ,( a+b) ` are in AP.
(ii) ` a^(2) (b+c) , b^(2)(c+a) ,c^(2) (a+b)` will be in AP.
if `b^(2) (c +a) -a^(2) (b+c) =c^(2) ( a+b) -b^(2) ( c+a)`
i.e, if ` c(b^(2) -a^(2)) + ab( b-a) =a (c^(2) -b^(2)) + bc ( c-b)`
i.e, if (b -a) (ab + bc+ ca) = ( c-a) ( ab + bc+ca)
i.e if ( b-a) = ( c -b)
i,e if 2b= ( a+c),which is ture by (i)
a,b,c are in AP ` Rightarrow a^(2) ( b+c) ,b^(2) (c+a) , c^(2) (a+b) ` are in AP.