If a,b,c,and d are in continued proportion,then prove (a-b)^3:(b-c)^3=a:d

Question

Using the properties of Proportions prove the sum

Answer 1

 
B = k1* A

C = k2* B = k2 * k1 * A

D = k3* C = k3*k2 *B = k3*k2*k1*A

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A - B = A(1-k1)

B - C = k1*A * (1-k2)

(A-B)^3 / (B-C)^3 = A^3*(1-k1)^3 /{ k1^3*A^3*(1-k2)^3}

= (1-k1)^3 / {k1^3 * (1-k2)^3 }

= [(1-k1)/k1]^3 * [1/(1-k2)]^3

= [ 1/k1 - 1]^3 * 1/(1-k2)^3


A/d = [(1-k1)/k1]^3 * [1/(1-k2)]^3

= [ 1/k1 - 1]^3 * 1/(1-k2)^3