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If the ${{n}^{th}}$ term of an AP is $6n+2$, then find the ${{5}^{th}}$ term of the AP is?

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Last updated date: 21st Jul 2024
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Answer
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Hint: To find the ${{5}^{th}}$ term of the AP we will use ${{n}^{th}}$ term of an A.P formula. Firstly we will write down the formula to find the ${{n}^{th}}$ term of an A.P then we will compare it by the ${{n}^{th}}$ term of the A.P given. Then we will get the value of the first term and the common difference of the A.P. Finally we will use the ${{n}^{th}}$ term of an A.P formula to get our ${{5}^{th}}$ term and desired answer.

Complete step-by-step solution:
It is given to us that ${{n}^{th}}$ term of the AP is as follows:
$6n+2$
So we can say that:
${{a}_{n}}=6n+2$……$\left( 1 \right)$
Now we know the formula to find ${{n}^{th}}$ term of the AP is as below:
${{a}_{n}}={{a}_{1}}+\left( n-1 \right)d$
Which when simplified is written as:
${{a}_{n}}={{a}_{1}}+dn-d$……$\left( 2 \right)$
On comparing coefficient of equation (1) and equation (2) we get,
By comparing coefficient of $n$
$d=6$…$\left( 3 \right)$
On comparing constant term,
${{a}_{1}}-d=2$
Put value from equation (3) above we get,
$\begin{align}
  & {{a}_{1}}-6=2 \\
 & \Rightarrow {{a}_{1}}=2+6 \\
\end{align}$
$\therefore {{a}_{1}}=8$…..$\left( 4 \right)$
Now as we have to find the ${{5}^{th}}$ term of the A.P so,
$n=5$…..$\left( 5 \right)$
 Put values from equation (3) (4) and (5) in equation (2) we get,
 $\begin{align}
  & {{a}_{5}}=8+6\times 5-6 \\
 & \Rightarrow {{a}_{5}}=8+30-6 \\
 & \therefore {{a}_{n}}=32 \\
\end{align}$
Hence ${{5}^{th}}$ term of the AP is 32.

Note: An A.P fully written as Arithmetic Progression is a sequence of numbers in a way that the difference between each consecutive number is constant i.e. there is common difference between each consecutive term. A finite portion of arithmetic progression is called a finite arithmetic progression. The sum of the members of a finite arithmetic progression is known as arithmetic series.