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 ?

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Answer 1 · 2018-08-11T14:15:19

Please see below.

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.