Derivation – Sum Of Arithmetic Series

 

Definition: Arithmetic Sequence is a sequence in which every term after the first term is obtained by adding a constant, called the common difference (d).

Sum of the arithmetic series, Sn, can be found by starting with the first term and successively adding the common difference.

We know an = a1 + (n – 1)d

Here, the first term is a1, second term is a1 + d, third term is a1 + 2d, etc.

Therefore,

Sn = a1 + (a1 + d) + (a1 + 2d) + … + [a1 + (n–1)d]

We could also start with the nth term and successively subtract the common difference. That is

Sn = an + (an – d) + (an– 2d) + … + [an – (n–1)d]

When you add these two equations together, we get,

Sn = a1 + (a1 + d) + (a1 + 2d) + … + [a1 + (n–1)d]

Sn = an + (an – d) + (an – 2d) + … + [an – (n–1)d]

 

2Sn = (a1 + an) + (a1 + an) + (a1 + an) + … + [a1 + an]

All the d terms are added out. So

2Sn = n (a1 + an)

Sn = n(a1 + an ) /  2

Sn = (n /2)(a1 + an )

When we substitute an = a1 + (n – 1)d into the last formula, we have

Sn = (n/ 2) [a1 + a1 + (n – 1)d]

Simplifying the above equation, we get

Sn = (n/2) [2 a1 + (n – 1)d]

These two formulas allow us to find the sum of an arithmetic series easily.