If a^2(b+c), b^2(c+a), c^2(a+b) are in A.P., show that either a,b,c are in A.P. or ab+bc+ca=0.

Asked by shaktidevgun | 19th Dec, 2015, 06:56: PM

Expert Answer:

G i v e n space t h a t space a squared open parentheses b plus c close parentheses comma space b squared open parentheses c plus a close parentheses space a n d space c squared open parentheses a plus b close parentheses space a r e space i n space A P. rightwards double arrow 2 b squared open parentheses c plus a close parentheses equals a squared open parentheses b plus c close parentheses plus c squared open parentheses a plus b close parentheses rightwards double arrow b squared open parentheses c plus a close parentheses minus a squared open parentheses b plus c close parentheses equals c squared open parentheses a plus b close parentheses minus b squared open parentheses c plus a close parentheses rightwards double arrow b squared c plus a b squared minus a squared b minus a squared c equals a c squared plus b c squared minus b squared c minus a b squared rightwards double arrow b squared c minus a squared c plus a b squared minus a squared b equals a c squared minus a b squared plus b c squared minus b squared c rightwards double arrow c open parentheses b squared minus a squared close parentheses plus a b open parentheses b minus a close parentheses equals a open parentheses c squared minus b squared close parentheses plus b c open parentheses c minus b close parentheses rightwards double arrow c open parentheses b minus a close parentheses open parentheses b plus a close parentheses plus a b open parentheses b minus a close parentheses equals a open parentheses c minus b close parentheses open parentheses c plus b close parentheses plus b c open parentheses c minus b close parentheses rightwards double arrow open parentheses b minus a close parentheses open square brackets c open parentheses b plus a close parentheses plus a b close square brackets equals open parentheses c minus b close parentheses open square brackets a open parentheses c plus b close parentheses plus b c close square brackets rightwards double arrow open parentheses b minus a close parentheses open square brackets c b plus a c plus a b close square brackets equals open parentheses c minus b close parentheses open square brackets a c plus a b plus b c close square brackets rightwards double arrow E i t h e r space open square brackets c b plus a c plus a b close square brackets equals 0 space o r space rightwards double arrow open parentheses b minus a close parentheses equals open parentheses c minus b close parentheses rightwards double arrow E i t h e r space open square brackets c b plus a c plus a b close square brackets equals 0 space o r space rightwards double arrow a comma b comma c space a r space e i n space A P

Answered by Vimala Ramamurthy | 21st Dec, 2015, 08:56: AM