The proof that the formula **S**n**=(a(1-r**^{n}**))/(1-r)** sums *n* terms of a geometric series is as follows, where *r* is the ratio of the series (which is between -1 and 1) and *a* is the first term in the series.

Sn=ar^{0}+ar^{1}+ar^{2}+...+ar^{n-2}+ar^{n-1}

(r)Sn=r(ar^{0}+ar^{1}+ar^{2}+...+ar^{n-1})

r*Sn=ar^{1}+ar^{2}+ar^{3}+...+ar^{n-1}+ar^{n}

a+r*Sn=a+ar^{1}+ar^{2}+ar^{3}+...+ar^{n-1}+ar^{n}

a+r*Sn=Sn+ar^{n}

Sn-r*Sn=a-ar^{n}

Sn(1-r)=a(1-r^{n}

Sn=(a(1-r^{n}))/(1-r)