Question

If a,b,c are in A.P then find a^2(b+c) , b^2(c+a) , c^2(a+b) are in A.P ?

Answer 1

Please see below.

Explanation:

As `a,b,c` are in A.P., we have `2b=a+c`

and hence `a^2(b+c)+c^2(a+b)`

= `a^2b+a^2c+c^2a+c^2b`

= `a^2b+ac(a+c)+bc^2`

= `a^2b+acxx2b+bc^2`

= `b(a^2+2ac+c^2)`

= `b(a+c)^2`

= `b*(2b)^2`

= `4b^3`

= `2b^2xx2b`

= `2b^2(a+c)`

i.e. `a^2(b+c)+c^2(a+b)=2b^2(a+c)`

or `c^2(a+b)-b^2(a+c)=b^2(a+c)-a^2(b+c)`

i.e. `a^2(b+c),b^2(c+a)` and `c^2(a+b)` are in A.P.