Formulas for Finding the Sum of an Arithmetic Series, Study notes of Calculus

Download Formulas for Finding the Sum of an Arithmetic Series and more Study notes Calculus in PDF only on Docsity! 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)2