If a^2(b+c) ,b^2(c+a) ,c^2(a+b) are in AP then prove that a,b,c are in AP

Asked by nk.pandit02 | 12th Mar, 2017, 07:39: PM

Expert Answer:

Dear student, the question should be
'If a^2(b+c) ,b^2(c+a) ,c^2(a+b) are in AP then prove that ab+bc+ac=0 or a,b,c are in AP' OR we  should be given that ab + bc + ac is not equal to 0.

begin mathsize 16px style Given space that space straight a squared open parentheses straight b plus straight c close parentheses comma space straight b squared open parentheses straight c plus straight a close parentheses space and space straight c squared open parentheses straight a plus straight b close parentheses space are space in space AP.
rightwards double arrow straight b squared open parentheses straight c plus straight a close parentheses minus straight a squared open parentheses straight b plus straight c close parentheses equals straight c squared open parentheses straight a plus straight b close parentheses minus straight b squared open parentheses straight c plus straight a close parentheses space
rightwards double arrow 2 straight b squared open parentheses straight c plus straight a close parentheses equals straight a squared open parentheses straight b plus straight c close parentheses plus straight c squared open parentheses straight a plus straight b close parentheses
rightwards double arrow straight b squared open parentheses straight c plus straight a close parentheses minus straight a squared open parentheses straight b plus straight c close parentheses equals straight c squared open parentheses straight a plus straight b close parentheses minus straight b squared open parentheses straight c plus straight a close parentheses
rightwards double arrow straight b squared straight c plus ab squared minus straight a squared straight b minus straight a squared straight c equals ac squared plus bc squared minus straight b squared straight c minus ab squared
rightwards double arrow straight b squared straight c minus straight a squared straight c plus ab squared minus straight a squared straight b equals ac squared minus ab squared plus bc squared minus straight b squared straight c
rightwards double arrow straight c open parentheses straight b squared minus straight a squared close parentheses plus ab open parentheses straight b minus straight a close parentheses equals straight a open parentheses straight c squared minus straight b squared close parentheses plus bc open parentheses straight c minus straight b close parentheses
rightwards double arrow straight c open parentheses straight b minus straight a close parentheses open parentheses straight b plus straight a close parentheses plus ab open parentheses straight b minus straight a close parentheses equals straight a open parentheses straight c minus straight b close parentheses open parentheses straight c plus straight b close parentheses plus bc open parentheses straight c minus straight b close parentheses
rightwards double arrow open parentheses straight b minus straight a close parentheses open square brackets straight c open parentheses straight b plus straight a close parentheses plus ab close square brackets equals open parentheses straight c minus straight b close parentheses open square brackets straight a open parentheses straight c plus straight b close parentheses plus bc close square brackets
rightwards double arrow open parentheses straight b minus straight a close parentheses open square brackets cb plus ac plus ab close square brackets equals open parentheses straight c minus straight b close parentheses open square brackets ac plus ab plus bc close square brackets
rightwards double arrow open parentheses straight b minus straight a close parentheses open square brackets cb plus ac plus ab close square brackets minus open parentheses straight c minus straight b close parentheses open square brackets ac plus ab plus bc close square brackets equals 0
rightwards double arrow open square brackets cb plus ac plus ab close square brackets open square brackets open parentheses straight b minus straight a close parentheses minus open parentheses straight c minus straight b close parentheses close square brackets equals 0
rightwards double arrow Either space
open square brackets cb plus ac plus ab close square brackets equals 0 space or space open parentheses straight b minus straight a close parentheses minus open parentheses straight c minus straight b close parentheses equals 0
rightwards double arrow cb plus ac plus ab equals 0 space or space open parentheses straight b minus straight a close parentheses equals open parentheses straight c minus straight b close parentheses
that space is comma space space ab plus bc plus ac equals 0 space or space straight a comma space straight b comma space straight c space arein space AP end style
 
Student, thank you for the appreciation.Smile

Answered by Rebecca Fernandes | 12th Mar, 2017, 07:52: PM

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