When the sequence of numbers (or a_{1}, a_{2}, a_{3},...) increases or decreases by a fixed quantity, then the sequence is in arithmetic progression (A.P.). The fixed quantity is called as *common difference*. For an AP, we define its first term as *a* and the common difference as *d*.
The general expression for an AP is: *a*, *a* + *d*, *a* + 2*d*, *a* + 3*d*,....

If T_{r} represents the gernal term of an AP, then
$ T_r = a + (r - 1)d $ where $ r \ \epsilon \ \{1, 2, 3,....n\} $

In an AP, the difference of any two consecutive terms is *d* and is given by:
$ d = T_r - T_{r-1} $

## Sum of *n* terms of an AP

Consider *n* terms of an AP with first term as a and common difference as d. Let S_{n} denote the sum of first *n* terms, then

$ S_n = a + (a+d) + (a + 2d) + .......+(a + (n - 1)d) $

```
$ S_n = \frac{n}{2}[2a + (n - 1)d] $
```

$ S_n = \frac{n}{2}(a + l) $ where $ l = a + (n - 1)d $

## Arithematic Mean

To understand the topic better, go to *Arithmetic mean*.
When three quantities are in AP, then the middle one is called as arithmetic mean of other two. If *a* and *b* are two numbers and A be the arithmetic mean of *a* and *b*, then *a*, A, *b* are in AP.

$ \Rightarrow A - a = b - A $

$ \Rightarrow A = \frac{a+b}{2} $

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