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**Hint:**An itemized collection of terms in which repetition of any sort of series are allowed is known as sequence.

Sum of n terms of such sequence means adding all the ‘n’ terms.

It could be done with various algebraic expressions

Here, \[{(a + b)^2} = {a^2} + {b^2} + 2ab\]

Further the concept of A.P & G.P series are used to solve such algebraic expressions

An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant. If the initial term of A.P is a, and common difference is ‘d’, then nth terms of sequence is

\[{a^n} = a + (n - 1)d,\] where n \[ = \] no. of terms.

Sum of A.P is given by \[{S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right],\] where n \[ = \] no. of terms, a \[ = \] first term, d\[ = \]common difference.

Geometric sequence has a common ratio.

The formula for the nth term is:

\[{a_n} = a{r^{n - 1}}\]

Where, \[{a_n} = \] nth term of the sequence

a\[ = \]first term of the sequence

r\[ = \]common ratio.

Sum of n terms of a G.P is denoted by \[{S_n}\]i.e.

\[{S_n} = \dfrac{{a\left( {{r^n} - 1} \right)}}{{(r - 1)}}\]

Where, a \[ = \]first term of the sequence.

r\[\, = \]common ratio.

n \[ = \] no. of terms in the sequence.

**Complete step-by-step answer:**

Let \[{S_n}\] denote the sum to n terms of the given sequence

Then,

\[{S_n} = {\left( {x + \dfrac{1}{x}} \right)^2} + {\left( {{x^2} + \dfrac{1}{{{x^2}}}} \right)^2} + {\left( {{x^3} + \dfrac{1}{{{x^2}}}} \right)^2} + ....\]

First, we need to find the general term of the sequence i.e. \[{\left( {{x^n} + \dfrac{1}{{{x^n}}}} \right)^2}\]

Now, \[{S_n} = {\left( {x + \dfrac{1}{x}} \right)^2} + {\left( {{x^2} + \dfrac{1}{{{x^2}}}} \right)^2} + {\left( {{x^3} + \dfrac{1}{{{x^2}}}} \right)^2} + ....{\left( {{x^n} + \dfrac{1}{{{x^n}}}} \right)^2}\]_____ (1).

Using identity \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\] throughout the sequence (1), we have:

\[{S_n} = \left[ {{x^2} + \dfrac{1}{{{x^2}}} + 2x \times x} \right] + \left[ {{x^4} + \dfrac{1}{{{x^4}}} + 2{x^2} \times \dfrac{1}{{{x^2}}}} \right] + \left[ {{x^6} + \dfrac{1}{{{x^6}}} + 2{x^6} \times \dfrac{1}{{{x^6}}}} \right] + ......\left[ {{x^{2n}} + \dfrac{1}{{{x^{2n}}}} + 2{x^n} \times \dfrac{1}{{{x^n}}}} \right]\]

Simplify the series ${x^2}$ is canceled out.

\[{S_n} = \left[ {{x^2} + \dfrac{1}{{{x^2}}} + 2} \right] + \left[ {{x^4} + \dfrac{1}{{{x^4}}} + 2} \right] + \left[ {{x^6} + \dfrac{1}{{{x^6}}} + 2} \right] + ......\left[ {{x^{2n}} + \dfrac{1}{{{x^{2n}}}} + 2} \right]\]

Combining term, we get:

\[{S_n} = \left[ {{x^2} + {x^4} + {x^6} + ..... + {x^{2n}}} \right] + \left[ {\dfrac{1}{{{x^2}}} + \dfrac{1}{{{x^4}}} + \dfrac{1}{{x{}^6}} + ......\dfrac{1}{{{x^{2n}}}}} \right] + \left[ {2 + 2 + .........2n} \right]\]. ______ (2).

We see that \[\left[ {{x^2} + {x^4} + {x^6} + ..... + {x^{2n}}} \right]\] is a G.P with common ratio \[{x^2}\]and a\[ = {x^2}\]

Sum of finite. G.P \[ = \dfrac{{a\left( {{r^n} - 1} \right)}}{{r - 1}}\]\[ = \dfrac{{{x^2}\left( {{{\left( {{x^2}} \right)}^n} - 1} \right)}}{{{x^2} - 1}}\]

\[\dfrac{{{x^2}\left[ {{x^{2n}} - 1} \right]}}{{{x^2} - 1}}\] _______(3)

Now,

\[\dfrac{1}{{{x^2}}} + \dfrac{1}{{{x^4}}} + ........ + \dfrac{1}{{{x^{2n}}}}\] is a G.P with a \[ = \dfrac{1}{{{x^2}}}\] and r \[ = \dfrac{1}{{{x^2}}}\]

Sum of finite G.P \[ = \dfrac{{a\left( {{r^n} - 1} \right)}}{{r - 1}} = \dfrac{{\dfrac{1}{{{x^2}}}\left( {{{\left( {\dfrac{1}{{{x^2}}}} \right)}^n} - 1} \right)}}{{\left( {\dfrac{1}{{{x^2}}}} \right) - 1}}\] ____________(4).

Substituting equation (3) & (4) in (2), we have:

\[{S_n} = {x^2}\left[ {\dfrac{{{x^{2n}} - 1}}{{{x^2} - 1}}} \right] + \dfrac{1}{{{x^2}}}\left[ {\dfrac{{{{\left( {\dfrac{1}{{{x^2}}}} \right)}^n} - 1}}{{\left( {\dfrac{1}{{{x^2}}}} \right) - 1}}} \right] + \left( {2 + .... + 2} \right)\]n times

n times \[2\] will be \[2n\]

\[{S_n} = {x^2}\left[ {\dfrac{{{x^{2n}} - 1}}{{{x^2} - 1}}} \right] + \dfrac{1}{{{x^2}}}\left[ {\dfrac{{{{\left( {\dfrac{1}{{{x^2}}}} \right)}^n} - 1}}{{\left( {\dfrac{1}{{{x^2}}}} \right) - 1}}} \right] + 2n\]

Simplify

\[{S_n} = {x^2}\left[ {\dfrac{{{x^{2n}} - 1}}{{{x^2} - 1}}} \right] + \dfrac{1}{{{x^2}}}\left[ {\dfrac{{\dfrac{{1 - {x^{2n}}}}{{{x^{2n}}}}}}{{\dfrac{{1 - {x^2}}}{{{x^2}}}}}} \right] + 2n\]

Common denominator is cancelled out.

\[{S_n} = {x^2}\left[ {\dfrac{{{x^{2n}} - 1}}{{{x^2} - 1}}} \right] + \dfrac{1}{{{x^{2n}}}}\left[ {\dfrac{{1 - {x^{2n}}}}{{1 - {x^2}}}} \right] + 2n\]

\[{S_n} = {x^2}\left[ {\dfrac{{{x^{2n}} - 1}}{{{x^2} - 1}}} \right] + \dfrac{1}{{{x^{2n}}}}\left[ {\dfrac{{ - \left( {{x^{2n}} - 1} \right)}}{{ - \left( {{x^2} - 1} \right)}}} \right] + 2n\]

Taking \[\left[ {\dfrac{{{x^{2n}} - 1}}{{{x^2} - 1}}} \right]\] common from first \[2\]terms

\[{S_n} = \left[ {\dfrac{{{x^{2n}} - 1}}{{{x^2} - 1}}} \right] + \left[ {{x^2} + \dfrac{1}{{{x^{2n}}}}} \right] + 2n\]

Hence, it is the required sum.

**Note:**In such a question all we need is to recognize the sequence whether it is a G.P or an A.P.

Also, as it clearly stated we need to find the sum of n terms i.e. we need to find the nth terms first. Arithmetic progression is of the form:- a-d, a, a+d, a+2d,.....

where a is the first term and

d is the common difference.

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