Sum of n Terms of Arithmetic Progression

The formula for the arithmetic progression sum is,

S_n = (n/2)[2a + (n – 1) × d]

S_n = (n/2)[a + l]

where,

• a is the First Term of Series
• l is the Last Term of Series
• n is the Number of Terms in Series

Derivation of Formula

Let ‘l’ denote the n_th term of the series and S_n be the sum of first n terms of AP a, (a+d), (a+2d), …., a+(n-1)d then,

S_n = a₁ + a₂ + a₃ + ….a_n-1 + a_n

S_n = a + (a + d) + (a + 2d) + …….. + (l – 2d) + (l – d) + l                      …(1)

Writing the series in reverse order, we get,

S_n = l + (l – d) + (l – 2d) + …….. + (a + 2d) + (a + d) + a                      …(2)

Adding equation (1) and (2),

2S_n = (a + l) + (a + l) + (a + l) + …….. + (a + l) + (a + l) + (a + l)

2S_n = n(a + l)

S_n = (n/2)(a + l)             …(3)

Hence, the formulae for finding the sum of a series is,

S_n = (n/2)(a + l) 

Replacing the last term l by the n^th term in equation 3 we get,

n^th term = a + (n – 1)d

S_n = (n/2)(a + a + (n – 1)d)

S_n = (n/2)(2a + (n – 1) x d)

Note: Consecutive terms in an Arithmetic Progression can also be represented as,

…….., a-3d , a-2d, a-d, a, a+d, a+2d, a+3d, ……..

Related Article:

• Summation Formula
• Sum of Natural Numbers
• Arithmetic Progression and Geometric Progression

Arithmetic progression(AP) also called an arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence

A progression is a sequence or series of numbers in which they are arranged in a particular order such that the relation between the consecutive terms of a series or sequence is always constant. In a progression, it is possible to obtain the n_th term of the series.

In mathematics, there are 3 types of progressions:

1. Arithmetic Progression (AP)
2. Geometric Progression (GP)
3. Harmonic Progression (HP)

let’s learn about AP in this article.