Question

The largest term in the sequence ${{a}_{n}}=\dfrac{{{n}^{2}}}{[{{n}^{3}}+200]}$ is given by
A. $\dfrac{529}{49}$
B. $\dfrac{8}{89}$
C. $\dfrac{49}{543}$
D. None of these

Answer 1

In this type of problem, firstly we have to assume the given sequence as a function, and then find the derivative of that function with respect to $x$ , after that derivative will be equal to zero and then find values where the derivative of the function will be maximum and then substitute those values in sequence and we will get our largest term.

Complete step by step answer:
A sequence can be defined as an arrangement of elements or numbers in some definite order. We can represent sequence as: ${{x}_{1}},{{x}_{2}},{{x}_{3}},.............{{x}_{n}}$ where $1,2,3$ represents the position of every number and here $n$ represents the ${{n}^{th}}$ term.
There are basically two types of sequences- Arithmetic sequence and geometric sequence.
Arithmetic Sequence can be defined as a list of elements or numbers with a definite pattern. In this sequence if we take any number and then subtract it by the previous number, then the difference that will come is always the same or constant. And the constant difference in all the pairs of successive numbers in a sequence known as the common difference and it is denoted by the letter $d$
If the common difference is positive, then the sequence is increasing, but when the difference is negative, then the sequence is decreasing.
Geometric Sequence can be defined as the collection of numbers in which each number is a constant multiple of the previous term. In this sequence if we take any number and then divide it by the previous number, then the ratio that we will get is always constant. In this sequence there is a common ratio between the successive terms and this common ratio is denoted by $r$
Now, as given in the question-
Let’s consider the given function:
${{a}_{n}}=f(x)=\dfrac{{{n}^{2}}}{[{{n}^{3}}+200]}$
$\Rightarrow f(x)=\dfrac{{{n}^{2}}}{[{{n}^{3}}+200]}$
Now, we will differentiate both the sides with respect to $x$ and we get as:
$\Rightarrow f'(x)=\dfrac{({{n}^{3}}+200)(2n)-({{n}^{2}})(3{{n}^{2}})}{{{({{n}^{3}}+200)}^{2}}}$
$\Rightarrow f'(x)=\dfrac{({{n}^{3}}+200)(2n)-({{n}^{2}})(3{{n}^{2}})}{{{({{n}^{3}}+200)}^{2}}}$
$\Rightarrow f'(x)=\dfrac{2{{n}^{4}}+400n-3{{n}^{4}}}{{{({{n}^{3}}+200)}^{2}}}$
$\Rightarrow f'(x)=\dfrac{n(-{{n}^{3}}+400)}{{{({{n}^{3}}+200)}^{2}}}$
Since, $n\in N$ $f'(x)\ne 0\forall n\in N$
For $n\le 7$ , $f'(x)>0$
For $n\ge 7$ , $f'(x)<0$
As from above conditions we can observe that at $n=7$ the given function is maximum. So we will get the largest term at $n=7$
Now, we will substitute the value $7$ in ${{a}_{n}}=\dfrac{{{n}^{2}}}{[{{n}^{3}}+200]}$
$\Rightarrow {{a}_{7}}=\dfrac{{{7}^{2}}}{[{{7}^{3}}+200]}$
$\Rightarrow {{a}_{7}}=\dfrac{49}{543}$
Therefore, ${{a}_{7}}=\dfrac{49}{543}$ is the largest term of the sequence.
Hence, the correct option is $3$

Note: Arithmetic Sequences are used in daily life for different purposes, such as determining the number of audience members an auditorium can hold, also used in algebra and geometry to solve many problems, can be used in calculating projected earnings from working for a company.