Sum of n Terms of Arithmetic Progression
The formula for the arithmetic progression sum is,
Sn = (n/2)[2a + (n – 1) × d]
Sn = (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 nth term of the series and Sn be the sum of first n terms of AP a, (a+d), (a+2d), …., a+(n-1)d then,
Sn = a1 + a2 + a3 + ….an-1 + an
Sn = a + (a + d) + (a + 2d) + …….. + (l – 2d) + (l – d) + l …(1)
Writing the series in reverse order, we get,
Sn = l + (l – d) + (l – 2d) + …….. + (a + 2d) + (a + d) + a …(2)
Adding equation (1) and (2),
2Sn = (a + l) + (a + l) + (a + l) + …….. + (a + l) + (a + l) + (a + l)
2Sn = n(a + l)
Sn = (n/2)(a + l) …(3)
Hence, the formulae for finding the sum of a series is,
Sn = (n/2)(a + l)
Replacing the last term l by the nth term in equation 3 we get,
nth term = a + (n – 1)d
Sn = (n/2)(a + a + (n – 1)d)
Sn = (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:
Arithmetic Progressions Class 10: NCERT Notes
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 nth term of the series.
In mathematics, there are 3 types of progressions:
- Arithmetic Progression (AP)
- Geometric Progression (GP)
- Harmonic Progression (HP)
let’s learn about AP in this article.
Table of Content
- Arithmetic Progressions
- Nth Term of an AP
- General Form of an AP
- Sum of n Terms of Arithmetic Progression
- Sample Problems on Arithmetic Progressions
- Practice Questions on Arithmetic Progression
- Arithmetic Progression-FAQs