Question

If a, b, c are in continued proportion, show: $a^\frac{1}{3} + b^\frac{1}{3}+ c^\frac{1}{3}= \frac{a}{b^2C^2} + \frac{b}{c2a^2} + \frac{c}{a^2b^2}$

Answer 1

Answer:

Step-by-step explanation:

concept:

a, b, c are said to be in continued proportion if a:b = b:c

given:

a, b, c are in continued proportion

then,

$\frac{a}{b}=\frac{b}{c}\\\\b^2=ac$

consider,

$\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2}\\\\=\frac{a}{(ac)c^2}+\frac{b}{(ac)^2}+\frac{c}{a^2(ac)}\\\\=\frac{a}{(ac^3}+\frac{b}{(b^2)^2}+\frac{c}{a^3c}\\\\=\frac{a}{ac^3}+\frac{b}{b^4}+\frac{c}{a^3c}\\\\=\frac{1}{c^3}+\frac{1}{b^3}+\frac{1}{a^3}\\\\=\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}$