Question

If ${S_n} = \dfrac{1}{{6.11}} + \dfrac{1}{{11.16}} + \dfrac{1}{{16.21}} + .......$ upto n then $6{S_n}$ is equals to
A. $\dfrac{{5n - 4}}{{5n + 6}}$
B. $\dfrac{n}{{5n + 6}}$
C. $\dfrac{{2n}}{{5n + 6}}$
D. $\dfrac{1}{{5n + 6}}$

Answer 1

Hint: We have to find the value of $6{S_n}$ using the value of ${S_n} = \dfrac{1}{{6.11}} + \dfrac{1}{{11.16}} + \dfrac{1}{{16.21}} + .......$. Firstly, we will simplify each term of ${S_n}$ to get a simplified pattern so that we can find the simplest value of ${S_n}$. Then we use that value to find $6{S_n}$. We simply multiply ${S_n}$ by $6$ to get the required answer.

Complete step by step Solution:
We have to find the value of $6{S_n}$
First we will simplify each term ${S_n} = \dfrac{1}{{6.11}} + \dfrac{1}{{11.16}} + \dfrac{1}{{16.21}} + .......$
$\dfrac{1}{{6.11}} = \dfrac{1}{5}\left( {\dfrac{1}{6} - \dfrac{1}{{11}}} \right)$
$\dfrac{1}{{11.16}} = \dfrac{1}{5}\left( {\dfrac{1}{{11}} - \dfrac{1}{{16}}} \right)$
$\dfrac{1}{{16.21}} = \dfrac{1}{5}\left( {\dfrac{1}{{16}} - \dfrac{1}{{21}}} \right)$
Similarly nth term
$\dfrac{1}{5}\left( {\dfrac{1}{{5n + 1}} - \dfrac{1}{{5n + 6}}} \right)$
${S_n} = \dfrac{1}{5}\left( {\dfrac{1}{6} - \dfrac{1}{{11}}} \right) + \dfrac{1}{5}\left( {\dfrac{1}{{11}} - \dfrac{1}{{16}}} \right) + \dfrac{1}{5}\left( {\dfrac{1}{{16}} - \dfrac{1}{{21}}} \right) + .......\dfrac{1}{5}\left( {\dfrac{1}{{5n + 1}} - \dfrac{1}{{5n + 6}}} \right)$
Taking $\dfrac{1}{5}$ common from previous
${S_n} = \dfrac{1}{5}\left( {\dfrac{1}{6} - \dfrac{1}{{11}} + \dfrac{1}{{11}} - \dfrac{1}{{16}} + \dfrac{1}{{16}} - \dfrac{1}{{21}} + .......\dfrac{1}{{5n + 1}} - \dfrac{1}{{5n + 6}}} \right)$

After simplifying, we get


${S_n} = \dfrac{1}{5}\left( {\dfrac{1}{6} - \dfrac{1}{{5n + 6}}} \right)$
${S_n} = \dfrac{1}{5}\left( {\dfrac{{5n + 6 - 6}}{{6(5n + 6)}}} \right)$
After solving,
${S_n} = \dfrac{1}{5}\left( {\dfrac{{5n}}{{6(5n + 6)}}} \right)$
${S_n} = \dfrac{n}{{6(5n + 6)}}$
Now, we multiply ${S_n}$ by 6 to get the final answer
$6{S_n} = \dfrac{{6n}}{{6(5n + 6)}}$
$6{S_n} = \dfrac{n}{{5n + 6}}$

Hence, the correct option is (d).

Additional Information:
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. . The study of series may be a major part of calculus and its generalization, mathematical analysis. Series area units are utilized in most areas of arithmetic, even for learning finite structures through generating functions. In additionally to their omnipresence in arithmetic, infinite series also are widely utilized in different quantitative disciplines like physics, engineering, statistics, and finance

Note: Students can make mistakes while calculating the value of ${S_n}$. They should pay attention while simplifying each term of ${S_n}$ to avoid any calculation error. Then after simplifying they should do further calculations carefully in order to get the correct answer. After that, they have carefully simplified the series ${S_n}$ so that they can get a simplified equation for further process. At last, they just have to multiply ${S_n}$ by 6.