Derivation – Sum of Arithmetic Series
Arithmetic Sequence is a sequence in which every term after the first is obtained
by adding a constant, called the common difference (d). To find the nth term of a
an arithmetic sequence, we know an = a1 + (n – 1)d
The first term is a1, second term is a1 + d, third term is a1 + 2d, etc
This leads up to finding the sum of the arithmetic series, Sn, by starting with the
first term and successively adding the common difference.
1st 2nd 3rd nth
Sn = a1 + (a1 + d) + (a1 + 2d) + … + [a1 + (n–1)d]
We could have also started with the nth term and successively subtracted the
common difference, so
Sn = an + (an – d) + (an– 2d) + … + [an – (n–1)d]
You could find the sum of the arithmetic sequence either way.
However, if you looked at that, you might see that if you added those two
equations together, terms add out.
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]
Notice all the d terms added out. So
2Sn = n (a1 + an)
Sn =n(a1+an)
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