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ICSE Class 10 Answered

If a,b,c are in continued proportion  Prove that  a^3+b^3+c^3 / a^2 b^2 c^2 = 1/a^3 + 1/b^3 +1/c^3
Asked by ujjwalkumarsingh2004 | 06 Jul, 2019, 04:59: PM
answered-by-expert Expert Answer
a, b and c are in the continued proportion
a/b = b/c = k
a = bk and b = ck then a = ck2
begin mathsize 16px style LHS space equals space fraction numerator straight a cubed plus straight b cubed plus straight c cubed over denominator straight a squared straight b squared straight c squared end fraction
equals fraction numerator straight c cubed straight k to the power of 6 plus straight c cubed straight k cubed plus straight c cubed over denominator straight c squared straight k to the power of 4 cross times straight c squared straight k squared cross times straight c squared end fraction
equals fraction numerator straight c cubed open parentheses straight k to the power of 6 plus straight k cubed plus 1 close parentheses over denominator straight c to the power of 6 straight k to the power of 6 end fraction
equals fraction numerator straight k to the power of 6 plus straight k cubed plus 1 over denominator straight c cubed straight k to the power of 6 end fraction space..... left parenthesis straight i right parenthesis
RHS space equals space 1 over straight a cubed plus 1 over straight b cubed plus 1 over straight c cubed
equals fraction numerator 1 over denominator straight c cubed straight k to the power of 6 end fraction plus fraction numerator 1 over denominator straight c cubed straight k cubed end fraction plus 1 over straight c cubed
equals 1 over straight c cubed open parentheses 1 over straight k to the power of 6 plus 1 over straight k cubed plus 1 close parentheses
equals fraction numerator straight k to the power of 6 plus straight k cubed plus 1 over denominator straight c cubed straight k to the power of 6 end fraction space space..... left parenthesis ii right parenthesis
From space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis
we space get space fraction numerator straight a cubed plus straight b cubed plus straight c cubed over denominator straight a squared straight b squared straight c squared end fraction equals fraction numerator straight k to the power of 6 plus straight k cubed plus 1 over denominator straight c cubed straight k to the power of 6 end fraction space if space straight a comma space straight b space and space straight c space are space in space the space conmtinued space proportion. end style
Answered by Sneha shidid | 08 Jul, 2019, 10:31: AM

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