Question

If ${{T}_{n}}=5n+2$, find the value of ${{T}_{n+1}}$.

Answer 1

Hint: The given series is an arithmetic progression, substitute $n+1$ at the place of $n$ to get the answer. Another way of solving the question can be by substituting the values of $n$ and forming a series and then finding the $\left( n+1 \right)th$ term.

Complete step-by-step answer:
In this question, Arithmetic Progression also known as A.P. will be used. An A.P. is a sequence of numbers such that the difference of any two successive numbers is a constant called common difference of the arithmetic progression. An A.P. is generally represented as:

$a,\text{ }a+d,\text{ }a+2d,\text{ }a+3d,\text{ }.........$, where $a$ is the first term and $d$ is the common difference.
To find the $n\text{th}$ term we use ${{T}_{n}}=a+(n-1)d$

Now, we come to the question. We have been given a sequence of the form ${{T}_{n}}=5n+2$. From this we can find different terms of the series. For example:
$\begin{align}
  & {{T}_{1}}=5\times 1+2=7 \\
 & {{T}_{2}}=5\times 2+2=12 \\
 & {{T}_{3}}=5\times 3+2=17 \\
 & {{T}_{4}}=5\times 4+2=22 \\
\end{align}$

We can clearly see that the first term of the series is 7 and the common difference between successive series is 5. Now, we have to find ${{T}_{n+1}}$. So, just substitute $n+1$ at the place of $n$ in the sequence ${{T}_{n}}=5n+2$. Therefore, we get, ${{T}_{n+1}}=5\left( n+1 \right)+2=5n+5+2=5n+7$.

Note: We can check our answer by taking the difference of $\left( n+1 \right)th$ term and $nth$ term. The result will be 5 and that is the common difference of the series and hence, it can be considered as a proof that our answer is correct. So, simply substitute the required value at the place of $n$, for the required term.