An arithmetic progression is a sequence in which each term (after the first) is determined by adding a constant to the preceding term. This constant is called the common difference of the arithmetic progression. An arithmetic progression can be defined as follows:

The arithmetic progression { a_n } = a₁, a₂, a₃, ...., a_n ,
where n = 1, 2, 3, . . .
Its terms are determined by the equation:

a_n = a₁ + (n - 1)d, where

a₁ is the first term of the arithmetic progression
a_n is the nth term of the arithmetic progression
n is the term number
d is the common difference of the arithmetic progression

The sum of the first n terms of an arithmetic progression is
calculated as

S_n = n ( a₁ + a_n ) / 2

or

S_n = n ( 2a₁ + (n - 1)d ) / 2 where a_n = a₁ + (n - 1)d

EX. For the sequence { a_n } = 1, 3, 5, 7, 9, ..... where a_n = 2n - 1

a_n = 2n - 1 = 1 + 2n - 2 = 1 + 2(n-1)

The sequence { a_n } = 1, 3, 5, 7, 9, ..... is an arithmetic sequence with a₁ = 1 and d = 2. The 6th to 10th terms of this arithmetic progression are

a₆ = 1 + 2(6-1) = 1 + 10 = 11
a₇ = 1 + 2(7-1) = 1 + 12 = 13
a₈ = 1 + 2(8-1) = 1 + 14 = 15
a₉ = 1 + 2(9-1) = 1 + 16 = 17
a₁₀ = 1 + 2(10-1) = 1 + 18 = 19

The sum of the first n terms of the sequence { a_n } = 1, 3, 5, 7, 9,. . . is
S_n = n (2(1) + (n - 1)2) / 2 = n (2 + 2n - 2) / 2 = 2n² / 2 = n²

We can verify this for the first 5 terms:

S₁ = 1² = 1
S₂ = 2² = 1 + 3 = 4
S₃ = 3² = 1 + 3 + 5 = 9
S₄ = 4² = 1 + 3 + 5 + 7 = 16
S₅ = 5² = 1 + 3 + 5 + 7 + 9 = 25