Answer
Verified
430.2k+ views
Hint: An expression is given and it has four divisions. It has a division within a division with a division within a division. So first solve the lowest division and after solving it go to the next one. Solve the equation by putting x as it is and then substitute the value of x.
Complete step-by-step answer:
We are given an expression $ \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{x}}}}}}} $ . We have to find its value when the value of x is 2011
So the given expression has four divisions.
So the first division to be solved is $ 1 + \dfrac{1}{x} $
$ 1 + \dfrac{1}{x} = \dfrac{{x + 1}}{x} $
After substituting the division value, the given expression will become
$ \Rightarrow \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{x}}}}}}} = \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{\left( {\dfrac{{x + 1}}{x}} \right)}}}}}} = \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{x}{{x + 1}}}}}} $
The next division is $ 1 + \dfrac{x}{{x + 1}} $
$ 1 + \dfrac{x}{{x + 1}} = \dfrac{{\left( {x + 1} \right) + x}}{{x + 1}} = \dfrac{{2x + 1}}{{x + 1}} $
On substituting the above division value, we get
$ \Rightarrow \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{x}{{x + 1}}}}}} = \dfrac{1}{{1 + \dfrac{1}{{\left( {\dfrac{{2x + 1}}{{x + 1}}} \right)}}}} = \dfrac{1}{{\left( {1 + \dfrac{{x + 1}}{{2x + 1}}} \right)}} $
The next division to be solved is $ 1 + \dfrac{{x + 1}}{{2x + 1}} $
$ \Rightarrow 1 + \dfrac{{x + 1}}{{2x + 1}} = \dfrac{{\left( {2x + 1} \right) + x + 1}}{{2x + 1}} = \dfrac{{3x + 2}}{{2x + 1}} $
Substituting the above division value in the expression, we get
$ \Rightarrow \dfrac{1}{{\left( {1 + \dfrac{{x + 1}}{{2x + 1}}} \right)}} = \dfrac{1}{{\left( {\dfrac{{3x + 2}}{{2x + 1}}} \right)}} = \dfrac{{2x + 1}}{{3x + 2}} $
Therefore, the value of the expression $ \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{x}}}}}}} $ is $ \dfrac{{2x + 1}}{{3x + 2}} $
We are asked to find its value when x is equal to 2011.
So we are substituting 2011 in the place of x.
$
\Rightarrow \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{2011}}}}}}}} = \dfrac{{2\left( {2011} \right) + 1}}{{3\left( {2011} \right) + 2}} \\
= \dfrac{{4022 + 1}}{{6033 + 2}} \\
= \dfrac{{4023}}{{6035}} \\
$
Therefore, the value of $ \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{x}}}}}}} $ when x is equal to 2011 is $ \dfrac{{4023}}{{6035}} $ .
So, the correct answer is “ $ \dfrac{{4023}}{{6035}} $ ”.
Additional Information: If the denominator of a fraction is zero, then the value of that fraction is infinity.
Note: The reciprocal of a number is the fraction in which the numerator is 1 and the denominator is the number itself. The reciprocal of a reciprocal of a number is the number itself. That is what we have done in the above solution. Let a fraction be $ \dfrac{a}{b} $ , then the reciprocal of this fraction will be $ \dfrac{b}{a} $ . The numerator of the original fraction will become the denominator in its reciprocal and the denominator of the original fraction will become the numerator in its reciprocal.
Complete step-by-step answer:
We are given an expression $ \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{x}}}}}}} $ . We have to find its value when the value of x is 2011
So the given expression has four divisions.
So the first division to be solved is $ 1 + \dfrac{1}{x} $
$ 1 + \dfrac{1}{x} = \dfrac{{x + 1}}{x} $
After substituting the division value, the given expression will become
$ \Rightarrow \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{x}}}}}}} = \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{\left( {\dfrac{{x + 1}}{x}} \right)}}}}}} = \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{x}{{x + 1}}}}}} $
The next division is $ 1 + \dfrac{x}{{x + 1}} $
$ 1 + \dfrac{x}{{x + 1}} = \dfrac{{\left( {x + 1} \right) + x}}{{x + 1}} = \dfrac{{2x + 1}}{{x + 1}} $
On substituting the above division value, we get
$ \Rightarrow \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{x}{{x + 1}}}}}} = \dfrac{1}{{1 + \dfrac{1}{{\left( {\dfrac{{2x + 1}}{{x + 1}}} \right)}}}} = \dfrac{1}{{\left( {1 + \dfrac{{x + 1}}{{2x + 1}}} \right)}} $
The next division to be solved is $ 1 + \dfrac{{x + 1}}{{2x + 1}} $
$ \Rightarrow 1 + \dfrac{{x + 1}}{{2x + 1}} = \dfrac{{\left( {2x + 1} \right) + x + 1}}{{2x + 1}} = \dfrac{{3x + 2}}{{2x + 1}} $
Substituting the above division value in the expression, we get
$ \Rightarrow \dfrac{1}{{\left( {1 + \dfrac{{x + 1}}{{2x + 1}}} \right)}} = \dfrac{1}{{\left( {\dfrac{{3x + 2}}{{2x + 1}}} \right)}} = \dfrac{{2x + 1}}{{3x + 2}} $
Therefore, the value of the expression $ \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{x}}}}}}} $ is $ \dfrac{{2x + 1}}{{3x + 2}} $
We are asked to find its value when x is equal to 2011.
So we are substituting 2011 in the place of x.
$
\Rightarrow \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{2011}}}}}}}} = \dfrac{{2\left( {2011} \right) + 1}}{{3\left( {2011} \right) + 2}} \\
= \dfrac{{4022 + 1}}{{6033 + 2}} \\
= \dfrac{{4023}}{{6035}} \\
$
Therefore, the value of $ \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{{1 + \dfrac{1}{x}}}}}}} $ when x is equal to 2011 is $ \dfrac{{4023}}{{6035}} $ .
So, the correct answer is “ $ \dfrac{{4023}}{{6035}} $ ”.
Additional Information: If the denominator of a fraction is zero, then the value of that fraction is infinity.
Note: The reciprocal of a number is the fraction in which the numerator is 1 and the denominator is the number itself. The reciprocal of a reciprocal of a number is the number itself. That is what we have done in the above solution. Let a fraction be $ \dfrac{a}{b} $ , then the reciprocal of this fraction will be $ \dfrac{b}{a} $ . The numerator of the original fraction will become the denominator in its reciprocal and the denominator of the original fraction will become the numerator in its reciprocal.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE