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

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

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

${\displaystyle S_n = a + (a+d) + (a + 2d) + .......+(a + (n - 1)d)}$

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

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

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.

${\displaystyle \Rightarrow A - a = b - A }$

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

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