# Derivation of Sum of Arithmetic Progression

**Arithmetic Progression, AP**

Definition

*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**

$S = a_1 + a_2 + a_3 + a_4 + ... + a_n$

$S = a_1 + (a_1 + d) + (a_1 + 2d) + (a_1 + 3d) + ... + [ \, a_1 + (n - 1)d \, ]$ ← Eq. (1)

$S = a_n + a_{n - 1} + a_{n - 2} + a_{n - 3} + ... + a_1$

$S = a_n + (a_n - d) + (a_n - 2d) + (a_n - 3d) + ... + [ \, a_n - (n - 1)d \, ]$ ← Eq. (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

$S = \dfrac{n}{2} \Big\{ \, a_1 + \Big[ \, a_1 + (n - 1)d \, \Big] \, \Big\}$

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