Arithmetic Progression (A.P.)

In a sequence, if the difference between any term and its preceding term ($t_{n+1}-t_n$) is constant, the sequence is called an Arithmetic Progression (A.P.).

Consider the sequences: (1) $2,5,8,11,14,\ldots$ (2) $4,10,16,22,28,\ldots$ (3) $4,16,64,256,\ldots$ (4) $\dfrac15,\dfrac1{25},\dfrac1{125},\ldots$ (5) $-3,2,7,12,17,\ldots$. Sequences (1), (2) and (5) are A.P.s, but the terms of (3) and (4) are not in A.P., since the difference between their consecutive terms is not constant.

If $t_1,t_2,t_3,\ldots,t_n$ are in A.P. then $t_{n+1}-t_n=d$ is constant for all $n$; $d$ is called the common difference. The sequence can then be written $a, a+d, a+2d, \ldots$ with first term $a$, so its $n$th term is $t_n=a+(n-1)d$, and the sum of the first $n$ terms is $S_n=t_1+t_2+\cdots+t_n=\dfrac{n}{2}[2a+(n-1)d]$. …