Geometric Series
Geometric Series
- \(S_{n}=a+ar+ar^2+ar^3+\dots+ar^{n-1}\)
- \(rS_{n}=ar+ar^2+ar^3+\dots+ar^n\)
- \(rS_{n}-S_{n}=\cancel{ ar+ar^2+ar^3\dots+ }ar^n-a\cancel{ -ar-ar^2-ar^3\dots-ar^{n-1} }=ar^n-a\)
- \(S_{n}(r-1)=a(r^n-1)\)
- \(S_{n}=\frac{a(r^n-1)}{r-1}\)
- or
- \(S_{n}=\frac{a(1-r^n)}{1-r}\)