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Permutations And Combinations

Question
CBSEENMA11013042

If a, b, c are in A.P. then prove that left parenthesis straight a minus straight c right parenthesis squared space equals space 4 left parenthesis straight b squared minus ac right parenthesis

Solution

Since a, b, c are in A.P.

∴           b - a = c - b      or    2b = a + c        or      straight b space equals space fraction numerator straight a plus straight c over denominator 2 end fraction                    ...(i)
left parenthesis straight a minus straight c right parenthesis squared space equals space 4 left parenthesis straight b squared minus ac right parenthesis          if            WiredFaculty
                                                                                            [ By using (i)]
if           straight a squared plus straight c squared minus 2 ac space equals space left parenthesis straight a plus straight c right parenthesis squared minus 4 ac  or  if  straight a squared plus straight c squared minus 2 ac space equals space straight a squared plus straight c squared plus 2 ac minus 4 ac
if              - 2ac = -2ac which is true

∴                left parenthesis straight a minus straight c right parenthesis squared space equals space 4 left parenthesis straight b squared minus ac right parenthesis

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