Geometric Progressions
Geometric Progressions
A geometric progression has a constant non-zero ratio between consecutive terms.
| Geometric Progression | |
|---|---|
| Notation | First term: \(a\) Common ratio: \(r\) |
| nth term | \[u_n=ar^{n-1}\] |
| Sum, \(S_n\) | \[S_n=\frac{a(r^n-1)}{r-1}\quad\text{or}\quad S_n=\frac{a(1-r^n)}{1-r}\] |
| Sum to infinity, \(S_\infty\) | The series converges for \(\lvert r\rvert<1\). \[S_\infty=\frac{a}{1-r}\] |
| Proving | \[\frac{u_n}{u_{n-1}}=\text{constant}\] (common ratio) |
| Mean | If \(x,y,z\) are consecutive terms: \[\frac{y}{x}=\frac{z}{y}\quad\Rightarrow\quad y=\pm\sqrt{xz}\] |
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