Real Numebrs Chapter 5 AP
1. Arithmetic Progression:
An arithmetic progression (AP) is a list of numbers in which each term is obtained by adding a fixed number $d$ to the preceding term, except the first term. The fixed number $d$ is called the common difference.
2.General Form of an AP:
The general form of an AP is $a, a+d, a+2 d, a+3 d, \ldots$
3.. A given list of numbers $a_{1}, a_{2}, a_{3}, \ldots$ is an AP if the differences $a_{2}-a_{1}, a_{3}-a_{2}$, $a_{4}-a_{3}, \ldots$, give the same value, i.e., if $a_{k+1}-a_{k}$ is the same for different values of $k$.
4. General Term of an AP
In an AP with first term $a$ and common difference $d$, the $n$th term (or the general term) is given by $a_{n}=a+(n-1) d$.
5. The sum of the first $n$ terms of an AP :
$$
\mathrm{S}=\frac{n}{2}[2 a+(n-1) d]
$$
6. If $l$ is the last term of the finite AP, say the $n$th term, then the sum of all terms of the AP is given by :
$$
\mathrm{S}=\frac{n}{2}(a+l)
$$