if a, b, c are in G.P., prove that $$a^2b^2c^2 [\dfrac{1}{a^3} + \dfrac{1}{b^3} + \dfrac{1}{c^3}] = a^3 + b^3 + c^3$$

Asked by Aaryan | 1 year ago |  87

##### Solution :-

Given that a, b, c are in GP.

By using the property of geometric mean,

b2 = ac

Let us consider LHS: $$a^2b^2c^2 [\dfrac{1}{a^3} + \dfrac{1}{b^3} + \dfrac{1}{c^3}]$$

$$\dfrac{ (ac)c^2}{a} + \dfrac{(b^2)^2}{b} + \dfrac{a^2(ac)}{c}$$ [by substituting the b2 = ac]

$$\dfrac{ac^3}{a} + \dfrac{b^4}{b} +\dfrac{ a^3c}{c}$$

c3 + b3 + a3 = RHS

LHS = RHS

Hence proved.

Answered by Aaryan | 1 year ago

### Related Questions

#### Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.

Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.

#### Find the two numbers whose A.M. is 25 and GM is 20.

Find the two numbers whose A.M. is 25 and GM is 20.

#### If a is the G.M. of 2 and 1/4 find a.

If a is the G.M. of 2 and $$\dfrac{1}{4}$$ find a.

#### Find the geometric means of the following pairs of numbers

Find the geometric means of the following pairs of numbers:

(i) 2 and 8

(ii) a3b and ab3

(iii) –8 and –2

#### Insert 5 geometric means between 32/9 and 81/2.

Insert 5 geometric means between $$\dfrac{32}{9}$$ and $$\dfrac{81}{2}$$.