Question

If ${x_i} > 0,i = 1,2,3....n$ then $\left( {{x_1} + {x_2} + .....{x_n}} \right)\left( {\dfrac{1}{{{x_1}}} + \dfrac{1}{{{x_2}}} + ..... + \dfrac{1}{{{x_n}}}} \right)$ is equal to
$\eqalign{
  & 1){n^2} \cr
  & 2) \geqslant {n^2} \cr
  & 3) \leqslant {n^2} \cr} $
4) none of these

Answer 1

Hint: The given question has an Arithmetic Progression and a Harmonic Progression. We need to multiply these two. The answers are in terms of $n$, so we need to find out the meaning of these progressions. Therefore, we will go ahead by finding out the Arithmetic mean and the Harmonic mean to get to the final answer.
The formulas used to solve the question are:
For Arithmetic Progression:
${a_n} = {a_1} + \left( {n - 1} \right)d$
Where, ${a_n}$is the ${n^{th}}$term of the sequence
${a_1}$is the ${1^{st}}$term of the sequence
$d$is the common difference between two terms in the sequence.
Arithmetic mean:
$A = \dfrac{{\left( {{x_1} + {x_2} + ..... + {x_n}} \right)}}{n}$
For Harmonic Progression:
$\dfrac{1}{{{h_n}}} = \dfrac{1}{{{h_1}}} + \left( {n - 1} \right)d$
This is very similar to AP, but is a reciprocal.
Harmonic mean:
$H = \dfrac{n}{{\left( {\dfrac{1}{{{x_1}}} + \dfrac{1}{{{x_2}}} + .... + \dfrac{1}{{{x_n}}}} \right)}}$

Complete step-by-step solution:
Let us consider the Arithmetic and Harmonic Means,
$A = \dfrac{{\left( {{x_1} + {x_2} + ..... + {x_n}} \right)}}{n}$
$H = \dfrac{n}{{\left( {\dfrac{1}{{{x_1}}} + \dfrac{1}{{{x_2}}} + .... + \dfrac{1}{{{x_n}}}} \right)}}$
We can say that, $A \geqslant H$from comparing the above equations.
That is, $\dfrac{{\left( {{x_1} + {x_2} + ..... + {x_n}} \right)}}{n} \geqslant \dfrac{n}{{\left( {\dfrac{1}{{{x_1}}} + \dfrac{1}{{{x_2}}} + .... + \dfrac{1}{{{x_n}}}} \right)}}$
By rearranging the equation, we get
$\left( {{x_1} + {x_2} + ..... + {x_n}} \right)\left( {\dfrac{1}{{{x_1}}} + \dfrac{1}{{{x_2}}} + .... + \dfrac{1}{{{x_n}}}} \right) \geqslant {n^2}$
Therefore, the final answer is $ \geqslant {n^2}$.
Hence, option (2) is the correct answer.

Additional Information:
Arithmetic progression is a sequence of numbers that have a common difference between the consecutive terms which is a constant. A Harmonic Progression is very similar to an Arithmetic progression, but is the reciprocal of the Arithmetic Progression.


Note: Note that we use the arithmetic mean and Harmonic mean formulas here instead of using the normal formula. Since the question has the means mentioned, we go for those formulas. So, to identify what is required, we need to know all the formulas. There are no simplifications required, all we do is just rearrange the equations from LHS to RHS.