Permutations and Combinations

Permutations and Combinations

Question

If a, b, c are in A.P. then prove that

straight a cubed plus 4 straight b cubed plus straight c cubed space equals space 3 straight b left parenthesis straight a squared plus straight c squared right parenthesis

Answer

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

∴        b - a = c - b           or           2b = a + c     or           b = fraction numerator straight a plus straight c over denominator 2 end fraction       ...(i)
straight a cubed plus 4 straight b cubed plus straight c cubed equals 3 straight b left parenthesis straight a squared plus straight c squared right parenthesis
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#6 {main}</pre>                                 [BY using (i)]
if    straight a cubed plus fraction numerator 4 left parenthesis straight a plus straight c right parenthesis cubed over denominator 8 end fraction plus straight c cubed space equals space fraction numerator 3 left parenthesis straight a cubed plus ac squared plus ca squared plus straight c cubed right parenthesis over denominator 2 end fraction
if     fraction numerator 2 straight a cubed plus straight a cubed plus straight c cubed plus 3 straight a squared straight c plus 3 ac squared plus 2 straight c cubed over denominator 2 end fraction space equals space fraction numerator 3 left parenthesis straight a cubed plus ac squared plus ca squared plus straight c cubed right parenthesis over denominator 2 end fraction
if      3 straight a cubed plus 3 straight c cubed plus 3 straight a squared straight c plus 3 ac squared space equals space 3 straight a cubed plus 3 straight c cubed plus 3 straight a squared straight c plus 3 ac squared which is true

∴          straight a cubed plus 4 straight b cubed plus straight c cubed space equals space 3 straight b left parenthesis straight a squared plus straight c squared right parenthesis

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