Derivation of Sum of Arithmetic Progression

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algebra
derivation of formula
progression
sequence
series
arithmetic progression

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 an in terms of a1 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 $
...

$ a_n = a_1 + (n - 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 $
...

$ a_1 = a_n - (n - 1)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 \, ] $   →   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 \, ] $   →   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) $

$ S = \dfrac{n}{2}(a_1 + a_n) $

 

Substitute an = a1 + (n - 1)d to the above equation, we have
$ S = \dfrac{n}{2} \{ \, a_1 + [ \, a_1 + (n - 1)d \, ] \, \} $

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

 

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