When the sequence of numbers (or a1, a2, a3,...) 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 + 2d, a + 3d,....

If Tr 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 Sn 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}