Arithmetic Progression, AP

Definition

Arithmetic Progression (also called arithmetic sequence), is a sequence of numbers such that the difference between any two consecutive terms is constant. Each term therefore in an arithmetic progression will increase or decrease at a constant value called the common difference, d.

Examples of arithmetic progression are:

• 2, 5, 8, 11,... common difference = 3
• 23, 19, 15, 11,... common difference = -4

Derivation of Formulas

Let
d = common difference
a_1 = first term
a_2 = second term
a_3 = third term
...
a_m = mth term or any term before a_n
...
a_n = nth term or last term

d = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 and so on.

Derivation for a_n in terms of a_1 and d

a_1 = a_1
a_2 = a_1 + d
a_3 = a_2 + d = (a_1 + d) + d = a_1 + 2d
a_4 = a_3 + d = (a_1 + 2d) + d = a_1 + 3d
a_5 = a_4 + d = (a_1 + 3d) + d = a_1 + 4d
...
a_m = a_1 + (m - 1)d
...

In similar manner
a_n = a_n
a_{n - 1} = a_n - d
a_{n - 2} = a_{n - 1} - d = (a_n - d) - d = a_n - 2d
a_{n - 3} = a_{n - 2} - d = (a_n - 2d) - d = a_n - 3d
a_{n - 4} = a_{n - 3} - d = (a_n - 3d) - d = a_n - 4d
...
a_m = a_n - (n - m)d
...

Derivation for the Sum of Arithmetic Progression, S

S = a_1 + a_2 + a_3 + a_4 + ... + a_n
S = a_1 + (a_1 + d) + (a_1 + 2d) + (a_1 + 3d) + ... + [ \, a_1 + (n - 1)d \, ] \, \to \, Equation (1)

S = a_n + a_{n - 1} + a_{n - 2} + a_{n - 3} + ... + a_1
S = a_n + (a_n - d) + (a_n - 2d) + (a_n - 3d) + ... + [ \, a_n - (n - 1)d \, ] \, \to \, Equation (2)

Add Equations (1) and (2)
2S = (a_1 + a_n) + (a_1 + a_n) + (a_1 + a_n) + (a_1 + a_n) + ... + (a_1 + a_n)
2S = n(a_1 + a_n)

Substitute a_n = a_1 + (n - 1)d to the above equation, we have
S = \dfrac{n}{2} \{ \, a_1 + [ \, a_1 + (n - 1)d \, ] \, \}