If $A, B, C, D, E$ are In Continued Proportion. Prove that $A : E = A^{4} : B^{4} .$

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Solution

Prove $A : E = A^{4} : B^{4} .$
Given: $A, B, C, D, E$ are In Continued Proportion.
Thus, $\frac{A}{B} = \frac{B}{C} = \frac{C}{D} = \frac{D}{E} = K$
So, we can write, $D = E K$
$C = D K \Rightarrow C = E K^{2}$
$B = C K \Rightarrow B = E K^{3}$
$A = B K \Rightarrow A = E K^{4}$
We want to prove $A : E = A^{4} : B^{4} .$
Taking the left-hand side:
$\frac{A}{E} = \frac{E K^{4}}{E} \Rightarrow \frac{A}{E} = K^{4}$ $\{A = E K^{4}\}$
Taking the right-hand side:
$\frac{A^{4}}{B^{4}} = {(\frac{A}{B})}^{4} \Rightarrow \frac{A^{4}}{B^{4}} = K^{4}$ $\{\frac{A}{B} = K\}$
L.H.S $=$ R.H.S
Hence, proved that $A : E = A^{4} : B^{4} .$

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