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

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.

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