If b+c-a/a, c+a-b/b, a+b-c/c are in AP show that 1/a, 1/b, 1/c are in AP provided a+b+c not equal to the zero 

Asked by nk.pandit02 | 12th Mar, 2017, 01:36: PM

Expert Answer:

begin mathsize 16px style Consider comma
fraction numerator straight b plus straight c minus straight a over denominator straight a end fraction equals straight b over straight a plus straight c over straight a minus 1
fraction numerator straight c plus straight a minus straight b over denominator straight b end fraction equals straight c over straight b plus straight a over straight b minus 1
fraction numerator straight a plus straight b minus straight c over denominator straight c end fraction equals straight a over straight c plus straight b over straight c minus 1
Since space fraction numerator straight b plus straight c minus straight a over denominator straight a end fraction comma space fraction numerator straight c plus straight a minus straight b over denominator straight b end fraction comma space fraction numerator straight a plus straight b minus straight c over denominator straight c end fraction space are space in space AP
rightwards double arrow straight b over straight a plus straight c over straight a minus 1 comma space straight c over straight b plus straight a over straight b minus 1 comma space straight a over straight c plus straight b over straight c minus 1 space are space in space AP
rightwards double arrow straight b over straight a plus straight c over straight a comma space straight c over straight b plus straight a over straight b comma space straight a over straight c plus straight b over straight c space are space in space AP
rightwards double arrow fraction numerator straight b plus straight c over denominator straight a end fraction comma space fraction numerator straight a plus straight c over denominator straight b end fraction comma space fraction numerator straight a plus straight b over denominator straight c end fraction space space are space in space AP
rightwards double arrow fraction numerator straight b plus straight c plus straight a minus straight a over denominator straight a end fraction comma space fraction numerator straight a plus straight c plus straight b minus straight b over denominator straight b end fraction comma space fraction numerator straight a plus straight b plus straight c minus straight c over denominator straight c end fraction space space are space in space AP
rightwards double arrow fraction numerator straight b plus straight c plus straight a over denominator straight a end fraction minus 1 comma space fraction numerator straight a plus straight c plus straight b over denominator straight b end fraction minus 1 comma space fraction numerator straight a plus straight b plus straight c over denominator straight c end fraction minus 1 space space are space in space AP
rightwards double arrow fraction numerator straight b plus straight c plus straight a over denominator straight a end fraction comma space fraction numerator straight a plus straight c plus straight b over denominator straight b end fraction comma space fraction numerator straight a plus straight b plus straight c over denominator straight c end fraction space space are space in space AP
space Since space straight a plus straight b plus straight c not equal to 0 comma space divide space throughout space by space straight a plus straight b plus straight c
rightwards double arrow 1 over straight a comma space 1 over straight b comma space 1 over straight c space space are space in space AP end style

Answered by Rebecca Fernandes | 12th Mar, 2017, 02:25: PM

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