Answer

Verified

438k+ views

**Hint:**Here we form the first few terms using substitution of values of n in the function. Write the terms after breaking them using place values. Calculate the sum of the geometric progression formed and write the value of the terms required in the equation. Calculate the value of the equation and check which of the given options match the answer.

* A geometric progression is a sequence of terms where each next term can be calculated by multiplying the term with the common ratio. If a GP has terms \[a,{a_2},{a_3}......{a_n}\] having common ratio ‘r’ then sum of ‘n’ terms of GP is given by \[S = \dfrac{{a({r^n} - 1)}}{{(r - 1)}}\]

**Complete step-by-step solution:**

Since we have \[{f_n}(x) = xxx.....x\](n digits)

If we put the value of \[n = 1 \Rightarrow {f_1}(x) = x\]........................… (1)

If we put the value of \[n = 2 \Rightarrow {f_2}(x) = xx\]

We can break the value\[{f_2}(x)\]using ones and tens place.

\[\because xx = x + 10x\]

\[ \Rightarrow {f_2}(x) = x + 10x\].......................… (2)

If we put the value of \[n = 3 \Rightarrow {f_3}(x) = xxx\]

We can break the value\[{f_3}(x)\]using ones, tens and hundreds place.

\[\because xxx = x + 10x + 100x\]

i.e. \[xxx = x + 10x + {10^2}x\]

\[ \Rightarrow {f_3}(x) = x + 10x + {10^2}x\]........................… (3)

So we can write general equation for ‘n’

\[ \Rightarrow {f_n}(x) = x + 10x + {10^2}x + {......10^{n - 1}}x\]..................… (4)

Now we take ‘x’ common from all terms in RHS of the equation (4)

\[ \Rightarrow {f_n}(x) = x\left( {1 + 10 + {{10}^2} + {{......10}^{n - 1}}} \right)\].............… (5)

The terms in the bracket form a GP as common ratio between each term is 10 i.e. \[\dfrac{{10}}{1} = 10;\dfrac{{100}}{{10}} = 10;......\dfrac{{{{10}^{n - 1}}}}{{{{10}^{n - 2}}}} = 10\]

So, the GP has a first term as 1, common ratio as 10. Use the formula of sum of n terms of a GP

\[ \Rightarrow {S_n} = \dfrac{{1({{10}^n} - 1)}}{{(10 - 1)}}\]

\[ \Rightarrow {S_n} = \dfrac{{{{10}^n} - 1}}{9}\]

So the value of \[\left( {1 + 10 + {{10}^2} + {{......10}^{n - 1}}} \right)\] is \[\dfrac{{{{10}^n} - 1}}{9}\].

Substitute the value of \[\left( {1 + 10 + {{10}^2} + {{......10}^{n - 1}}} \right)\] in equation (5)

\[ \Rightarrow {f_n}(x) = \dfrac{{x\left( {{{10}^n} - 1} \right)}}{9}\]..................… (6)

Now we have to find the value of the equation\[{f_n}^2(3) + {f_n}(2)\].

Calculate \[{f_n}(3)\] and \[{f_n}(2)\] separately.

Substitute value of \[x = 3\]in equation (6)

\[ \Rightarrow {f_n}(3) = \dfrac{{3\left( {{{10}^n} - 1} \right)}}{9}\]

Cancel same terms from numerator and denominator

\[ \Rightarrow {f_n}(3) = \dfrac{{\left( {{{10}^n} - 1} \right)}}{3}\]

Square both sides of the equation

\[ \Rightarrow {f_n}^2(3) = \dfrac{{{{\left( {{{10}^n} - 1} \right)}^2}}}{{{3^2}}}\]

\[ \Rightarrow {f_n}^2(3) = \dfrac{{{{\left( {{{10}^n} - 1} \right)}^2}}}{9}\].................… (7)

Substitute value of \[x = 2\]in equation (6)

\[ \Rightarrow {f_n}(2) = \dfrac{{2\left( {{{10}^n} - 1} \right)}}{9}\]...................… (8)

Substitute values from equations (7) and (8) in the equation\[{f_n}^2(3) + {f_n}(2)\]

\[ \Rightarrow {f_n}^2(3) + {f_n}(2) = \dfrac{{{{\left( {{{10}^n} - 1} \right)}^2}}}{9} + \dfrac{{2\left( {{{10}^n} - 1} \right)}}{9}\]

Take \[\dfrac{{\left( {{{10}^n} - 1} \right)}}{9}\]common in RHS

\[ \Rightarrow {f_n}^2(3) + {f_n}(2) = \dfrac{{\left( {{{10}^n} - 1} \right)}}{9}\left( {\left( {{{10}^n} - 1} \right) + 2} \right)\]

Calculate the value in the bracket

\[ \Rightarrow {f_n}^2(3) + {f_n}(2) = \dfrac{{\left( {{{10}^n} - 1} \right)}}{9}\left( {{{10}^n} - 1 + 2} \right)\]

\[ \Rightarrow {f_n}^2(3) + {f_n}(2) = \dfrac{{\left( {{{10}^n} - 1} \right)}}{9}\left( {{{10}^n} + 1} \right)\]

Use the identity \[(a - b)(a + b) = {a^2} - {b^2}\]

\[ \Rightarrow {f_n}^2(3) + {f_n}(2) = \dfrac{{\left( {{{({{10}^n})}^2} - {{(1)}^2}} \right)}}{9}\]

\[ \Rightarrow {f_n}^2(3) + {f_n}(2) = \dfrac{{\left( {{{10}^{2n}} - 1} \right)}}{9}\] ……………..… (9)

Now we check for the options

Since the power of 10 in the answer is 2n, we will consider the terms having subscript 2n first

Calculate \[{f_{2n}}(1)\]

Since \[{f_n}(x) = \dfrac{{x\left( {{{10}^n} - 1} \right)}}{9}\]

\[ \Rightarrow {f_{2n}}(x) = \dfrac{{x\left( {{{10}^{2n}} - 1} \right)}}{9}\]

Put \[x = 1\]in the above equation

\[ \Rightarrow {f_{2n}}(1) = \dfrac{{1\left( {{{10}^{2n}} - 1} \right)}}{9}\]

\[ \Rightarrow {f_{2n}}(1) = \dfrac{{\left( {{{10}^{2n}} - 1} \right)}}{9}\].......................… (10)

Since RHS of equations (9) and (10) are equal then LHS of equations are also equal.

\[ \Rightarrow {f_n}^2(3) + {f_n}(2) = {f_{2n}}(1)\]

\[\therefore \]The value of \[{f_n}^2(3) + {f_n}(2)\] is equal to\[{f_{2n}}(1)\]

**\[\therefore \]Correct option is C.**

**Note:**Many students get confused while writing the initial values of the functions as they write \[xx = {x^2};xxx = {x^3}...\]. This is the wrong approach as we are given that the function gives us n digits of the same type, so numbers formed can be like 999, 44, 333333 etc.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How do you graph the function fx 4x class 9 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The only snake that builds a nest is a Krait b King class 11 biology CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Why is there a time difference of about 5 hours between class 10 social science CBSE

Which places in India experience sunrise first and class 9 social science CBSE