Answer

Verified

417k+ views

**Hint:**In this question, we used continued fraction expansion. And fraction is that in mathematics, a fraction is an expression obtained through an iterative process of representing verity because the sum of its integer part and therefore the reciprocal of another number, then writing this other number because the sum of its integer part and another reciprocal, and so on.

\[

{a_0} + \dfrac{1}{{{a_1} + \dfrac{1}{{{a_2} + \dfrac{1}{{}}}}}} \\

\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;. \\

\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;. \\

\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;. + \dfrac{1}{{{a_n}}} \\

\]

It is a finite continued fraction, where n is a non-negative integer, \[{a_0}\] is an integer, and \[{a_i}\] is a positive integer, for \[i = 1,.............n\].

It is generally assumed that the numerator of the entire fraction is \[1\]. If the arbitrary values and functions are utilized in place of one or more of the numerator or the integer in the denominators, the resulting expression may be a generalized continued fraction.

**Complete step by step answer:**

The number is \[55\].

The factor of \[55 = 5*11\] has no square factors, so \[\sqrt {55} \] can’t be simplified.

Then,

It is an irrational approximation; I will find a continued fraction expansion for \[\sqrt {55} \] then truncate it.

To find the simple continued fraction expansion of \[\sqrt n \], we use the following algorithm.

\[

{m_0} = 0 \\

{d_0} = 1 \\

{a_0} = \sqrt n \\

{m_{i + 1}} = {d_i}{a_i} - {m_i} \\

{d_{i + 1}} = \dfrac{{n - {m^2}_{i + 1}}}{{{d_i}}} \\

{a_{i + 1}} = \dfrac{{{a_0} + {m_{i + 1}}}}{{{d_{i + 1}}}} \\

\]

This algorithm stops when \[{a_i} = 2{a_0}\], making the end of the repeating part of the continued fraction.

Then, the continued fraction expansion is.

\[\left[ {{a_0};\;{a_1},\;{a_2},\;{a_3}........} \right] = {a_0} + \dfrac{1}{{{a_1} + \dfrac{1}{{{a_2} + \dfrac{1}{{{a_3} + ........}}}}}}\]

Next, in the question the value of \[n = 55\] and\[\left[ {\sqrt n } \right] = 7\], since \[{7^2} = 49 < 55 < 64 = {8^2}\].

So by using the continued fraction expansion:

\[

{m_0} = 0 \\

{d_0} = 1 \\

{a_0} = \left[ {\sqrt {55} } \right] = 7 \\

{m_1} = {d_0}{a_0} - {m_0} = 7 \\

{d_1} = \dfrac{{n - {m_1}^2}}{{{d_0}}} = \dfrac{{55 - {7^2}}}{1} = 6 \\

{a_1} = \left[ {\dfrac{{{a_0} + {m_1}}}{{{d_1}}}} \right] = \left[ {\dfrac{{7 + 7}}{6}} \right] = 2 \\

{m_2} = {d_1}{a_1} - {m_1} = 12 - 7 = 5 \\

{d_2} = \dfrac{{n - {m_2}^2}}{{{d_1}}} = \dfrac{{55 - 25}}{6} = 5 \\

{a_2} = \left[ {\dfrac{{{a_0} + {m_2}}}{{{d_2}}}} \right] = \left[ {\dfrac{{7 + 5}}{5}} \right] = 2 \\

{m_3} = {d_2}{a_2} - {m_2} = 10 - 5 = 5 \\

{d_3} = \dfrac{{n - {m_3}^2}}{{{d_2}}} = \dfrac{{55 - 25}}{5} = 6 \\

{a_3} = \left[ {\dfrac{{{a_0} + {m_3}}}{{{d_3}}}} \right] = \left[ {\dfrac{{7 + 5}}{6}} \right] = 2 \\

{m_4} = {d_3}{a_3} - {m_3} = 12 - 5 = 7 \\

{d_4} = \dfrac{{n - {m_4}^2}}{{{d_3}}} = \dfrac{{55 - 49}}{6} = 1 \\

{a_4} = \left[ {\dfrac{{{a_0} + {m_4}}}{{{d_4}}}} \right] = \left[ {\dfrac{{7 + 7}}{1}} \right] = 14 \\

\]

Having reached a value \[14\] which is twice the primary value \[7\], this is often the top of the repeating pattern of the fraction, and that we have:

\[\sqrt {55} = \left[ {7;\;2,\;2,\;1,\;14} \right]\]

The first economical approximation for \[\sqrt {55} \] is then:

\[

\Rightarrow \sqrt {55} \approx \left[ {7;2,2,1} \right] = 7 + \dfrac{1}{{2 + \dfrac{1}{{2 + \dfrac{1}{1}}}}} \\

= \dfrac{{52}}{7} = 7.4285 \\

\]

Then, we again used the repeated value.

\[

\Rightarrow \sqrt {55} = \left[ {7;2,2,1,14,2,2,1} \right] \\

\approx 7.42857142857 \\

\]

**Therefore the closer value of square root of \[55\] is:**

\[\therefore \sqrt {55} \approx 7.42857142857\]

\[\therefore \sqrt {55} \approx 7.42857142857\]

**Note:**

As we know that continued fraction is just another way of writing fraction. They have some interesting connections with a jigsaw puzzle problem about splitting a rectangle into squares etc. it is the simple method for finding the square root of a number which has no square factor.

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

Which are the Top 10 Largest Countries of the World?

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

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

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

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

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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