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Question
If a, b, c are in continued proportion, prove that: abc(a + b + c)3 = (ab + bc + ca)3
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Solution
As a, b, c, are in continued proportion
Let `a/b = b/c` = k
L.H.S. = abc(a + b + c)3
= ck2.ck.c[ck2 + ck + c]3
= c3k3[c(k2 + k + 1)]3
= c3k3.c3.(k2 + k + 1)3
= c6k3(k2 + k + 1)3
R.H.S. = (ab + bc + ca)3
= (ck2.ck + ck.c + c. ck2)3
= (c2k3 + c2k + c2k2)3
= (c2k3 + c2k2 + c2k)3
= [c2k(k2 + k + 1)]3
= c6k3(k + k + 1)3
∴ L.H.S. = R.H.S.
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