Mathematics
If a, b, c and d are in continued proportion, prove that ad(c2 + d2) = c3(b + d).
Ratio Proportion
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Answer
Since, a, b, c, d are in continued proportion.
(let).
c = dk, b = ck = (dk)k = dk2, a = bk = (dk2)k = dk3.
Substituting values in L.H.S. of the equation ad(c2 + d2) = c3(b + d), we get :
L.H.S = ad(c2 + d2)
= dk3.(d).[(dk)2 + d2]
= d2k3.[d2(k2 + 1)]
= d4k3(k2 + 1).
Substituting values in R.H.S. of the equation ad(c2 + d2) = c3(b + d), we get :
R.H.S = c3(b + d)
= (dk)3.(dk2 + d)
= d3k3[d(k2 + 1)]
= d4k3(k2 + 1).
Since, L.H.S = R.H.S
Hence, proved that ad(c2 + d2) = c3(b + d).
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