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In a certain series if ${{S}_{n}}=\dfrac{1}{6}n(n+1)(n+2)$, then find ${{S}_{2}}$ .

Last updated date: 21st Jul 2024
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Hint: In question we are given a series ${{S}_{n}}=\dfrac{1}{6}n(n+1)(n+2)$. Here, we have to find ${{S}_{2}}$. So to find ${{S}_{2}}$ substitute $n=2$. Simplify the equation. You will get the answer.

Complete step-by-step answer:
A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
For example, the sequence $2,6,18,....$ is a geometric progression with a common ratio $3$.
Similarly $10,5,2.5,......$ is a geometric sequence with common ratio $\dfrac{1}{2}$.
Examples of a geometric sequence are powers of a fixed number ${{r}^{k}}$, such as ${{2}^{k}}$ and ${{3}^{k}}$.
The general form of a geometric sequence is,
The ${{n}^{th}}$ term of a geometric sequence with initial value $a={{a}_{0}}$ and the common ratio $r$ is given by,
Such a geometric sequence also follows the recursive relation.
${{a}_{n}}={{a}_{^{n-1}}}r$ for every integer $n\ge 1$.
Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio.
The common ratio of a geometric sequence may be negative, resulting in an alternating sequence, with numbers alternating between positive and negative.
Arithmetic Progression (A.P) is a sequence of numbers in a particular order. If we observe in our regular lives, we come across progression quite often. For example, roll numbers of a class, days in a week, or months in a year. This pattern of series and sequences has been generalized in maths as progressions. Let us learn here AP definition, important terms such as common difference, the first term of the series, nth term and sum of nth term formulas along with solved questions based on them.
It is a mathematical sequence in which the difference between two consecutive terms is always a constant and it is abbreviated as A.P.
The fixed number that must be added to any term of an A.P to get the next term is known as the common difference of the A.P.
An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.
${{n}^{th}}$ term of A.P,
Now we know for the A.P sum of ${{n}^{th}}$ the term is,
$S=\dfrac{n}{2}\left( 2a+(n-1)d \right)$
$d=$ Common difference
$n=$ number of terms

In the question we have been given a series ${{S}_{n}}=\dfrac{1}{6}n(n+1)(n+2)$.
We have to find ${{S}_{2}}$, for that substitute $n=2$.
Simplifying we get,
  & {{S}_{2}}=\dfrac{1}{6}2(3)(4) \\
 & {{S}_{2}}=4 \\
So we get the value of ${{S}_{2}}$ is $4$.

Note: Read the question carefully. Your concept regarding A.P and G.P should be clear. Also, you must know the properties. Don’t make any mistake while simplifying. Do not miss any term while simplifying.