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} $
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