Permutations and Combinations

Permutations and Combinations

Question

If straight a squared comma space straight b squared comma space straight c squared are in A.P., then prove that fraction numerator straight a over denominator straight b plus straight c end fraction comma space fraction numerator straight b over denominator straight c plus straight a end fraction comma space fraction numerator straight c over denominator straight a plus straight b end fraction are in A.P.

Answer
fraction numerator straight a over denominator straight b plus straight c end fraction comma space fraction numerator straight b over denominator straight c plus straight a end fraction comma space fraction numerator straight c over denominator straight a plus straight b end fraction will be in A.P.
if fraction numerator straight a over denominator straight b plus straight c end fraction plus 1 comma space space fraction numerator straight b over denominator straight c plus straight a end fraction plus 1 comma space space fraction numerator straight c over denominator straight a plus straight b end fraction plus 1 will be in A.P.                (By adding 1 to each number)
if  fraction numerator straight a plus straight b plus straight c over denominator straight b plus straight c end fraction comma space space fraction numerator straight a plus straight b plus straight c over denominator straight c plus straight a end fraction comma space fraction numerator straight a plus straight b plus straight c over denominator straight a plus straight b end fraction will be in A.P.
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#6 {main}</pre> will be in A.P.                   (By dividing each number a + b + c)
if     space space space space fraction numerator 1 over denominator straight c plus straight a end fraction minus fraction numerator 1 over denominator straight b plus straight c end fraction space equals space fraction numerator 1 over denominator straight a plus straight b end fraction minus fraction numerator 1 over denominator straight c plus straight a end fraction  if    space space fraction numerator straight b plus straight c minus straight c minus straight a over denominator left parenthesis straight c plus straight a right parenthesis space left parenthesis straight b plus straight c right parenthesis end fraction space equals space fraction numerator straight c plus straight a minus straight a minus straight b over denominator left parenthesis straight a plus straight b right parenthesis space left parenthesis straight c plus straight a right parenthesis end fraction
if       fraction numerator straight b minus straight a over denominator straight c plus straight b end fraction space equals space fraction numerator straight c minus straight b over denominator straight b plus straight a end fraction                       if        (b - a) (b + a) = (c - b) (c + b)
if      space space space space straight b squared minus straight a squared space equals space straight c squared minus straight b squared
if        space space straight a squared comma space straight b squared comma space straight c squared
  are in A.P. which is true.
Hence,space space space space space fraction numerator straight a over denominator straight b plus straight c end fraction comma space space fraction numerator straight b over denominator straight c plus straight a end fraction comma space space fraction numerator straight c over denominator straight a plus straight b end fraction are in A.P.

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