Question

What is the ${{30}^{th}}$ term of the following sequence $2,8,14,20?$

Answer 1

To solve this question we need to know the concept of Arithmetic Progression. A series is said to be in A.P when the common difference between all the two terms are the same. We find the value of common difference by subtracting the two consecutive terms in the series. To find the ${{n}^{th}}$ term in a A.P series, the formula we use is, ${{T}_{n}}=a+\left( n-1 \right)d$.

Complete step by step answer:
To solve this question we need to know about the behaviour of the sequence, as per the question we need to find the $30^{th}$ term. As we can see that the sequence starts from $2$, so we need to find the term ${{30}^{th}}$. The above given sequence is in Arithmetic Progression, AP as the common difference of the term is $6$. Common difference is found as ${{\left( n+1 \right)}^{th}}-{{\left( n \right)}^{th}}$ . We may check by subtracting ${{3}^{rd}}$ from ${{4}^{th}}$ which is:
$\Rightarrow {{4}^{th}}-{{3}^{rd}}$
$\Rightarrow 20-14$
$\Rightarrow 6$
Similarly we will find the difference by subtracting ${{1}^{st}}$ term from ${{2}^{nd}}$
$\Rightarrow {{2}^{nd}}-{{1}^{st}}$
$\Rightarrow 8-2$
$\Rightarrow 6$
Since the difference between the consecutive terms is $6$, so the common difference is $1$.
We can find the ${{n}^{th}}$ term of a sequence by using the formula $a+\left( n-1 \right)d$ , here $''a''$ is the first term, $''n''$ is the number of terms and $''d''$ is the common difference of the AP. The values of these terms given in the question are$a=2,n=30,d=6$. Substituting these values in the same formula, we get:
$\Rightarrow {{T}_{n}}=a+\left( n-1 \right)d$
$\Rightarrow {{T}_{30}}=2+\left( 30-1 \right)\times 6$
Now using BODMAS to calculate the above expression we get:
$\Rightarrow {{T}_{30}}=2+\left( 29 \right)\times 6$
$\Rightarrow {{T}_{30}}=2+174$
$\Rightarrow {{T}_{30}}=176$

$\therefore$ The ${{30}^{th}}$ term of the sequence natural numbers is $2,8,14,20$ is $176$.

Note: When numbers are in a certain series or sequence finding a particular term becomes easier. So the first step for the calculation of these types of the series, try finding the relation between each term. If we are able to find a certain relation between the terms in the series formula can directly be applied in that situation.