Mathematics
If a, b and c are in continued proportion, prove that :
abc(a + b + c)3 = (ab + bc + ca)3.
Ratio Proportion
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Answer
Given,
a, b and c are in continued proportion.
∴ = k (let)
⇒ b = ck and a = bk = ck.k = ck2.
Substituting value of a and b in abc(a + b + c)3, we get :
⇒ ck2.ck.c(ck2 + ck + c)3
⇒ c3k3[c(k2 + k + 1)]3
⇒ c3k3.c3(k2 + k + 1)3
⇒ c6k3(k2 + k + 1)3.
Substituting value of a and b in (ab + bc + ca)3, we get :
⇒ (ck2.ck + ck.c + c.ck2)3
⇒ (c2k3 + c2k + c2k2)3
⇒ [(c2k)3(k2 + 1 + k)3]
⇒ c6.k3(k2 + k + 1)3.
Since, L.H.S. = R.H.S.
Hence, proved that abc(a + b + c)3 = (ab + bc + ca)3.
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