Question

Fill in the blanks in the following table, given a is the first term and, d is a common difference and ${{a}_{n}}$ the $n^{th}$ term of the AP
$a=7$, $d=3$, n=8, find ${{a}_{n}}$

Answer 1

Arithmetic progression is the series of numbers that have a pattern and the pattern is such that the series changes in the same way after the addition of a particular number. And the ${{n}^{th}}$ term of that series can be expressed in terms of a formula.

Complete step by step answer:
Each series in the arithmetic progression has a starting point and the starting point or number can be any number. Let us assume that the first term to be $a$. Now, as the series moves on, it is such that the difference between the second term and the first term is the same as the difference between the third term and the second term and that is the same as the difference between the fourth term and the third term and so on.
This number is called the common difference of the arithmetic progression. And let that be $d$. Now, each term has a ${{n}^{th}}$term and that can be easily expressed in terms of the common difference and the first term and $n$ itself.
So the ${{n}^{th}}$term of an A.P is
${{a}_{n}}=a+\left( n-1 \right)d$
Now, In the question, the first term, the common difference and the number $n$ is given and we are to find the ${{n}^{th}}$term.
Therefore, the answer is
${{a}_{n}}=7+\left( 8-1 \right)3$
Now If we simplify the answer comes out to be $28$.
That is ${{a}_{n}}$ is $28$.

Note: Apart from arithmetic progression, there are various other types of series that have their own formulas of the $n^{th}$ term. A very popular one is the Geometric progression which comes after the arithmetic progression.