| Pattern | \( n\)th | Sequence |
|---|
| \(e, \dfrac{e^{2}}{2}, \dfrac{e^{3}}{3}, \dfrac{e^{4}}{4},\ldots\) | \(a_{n}=\dfrac{e^{n}}{n}\) | \( \{ a_{n}\} =\left\{ \dfrac{e^{n}}{n}\right\}\) |
| \(1,~\dfrac{1}{3},~\dfrac{1}{9},~\dfrac{1}{27},\ldots\) | \(b_{n}=\dfrac{1}{3^{n-1}}\) | \( \{ b_{n}\} =\left\{ \dfrac{1}{3^{n-1}}\right\}\) |
| \(1,~3,~5,~7,\ldots\) | \(c_{n}=2n-1\) | \( \{ c_{n}\} = \{2n-1\}\) |
| \(1,~4,~9,~16,~25,\ldots\) | \(d_{n}=n^{2}\) | \( \{ d_{n}\} = \{n^{2}\}\) |
| \(1,~{-}\dfrac{1}{2},~\dfrac{1}{3},~{-}\dfrac{1}{4},~\dfrac{1}{5},\ldots\) | \(e_{n}= ( -1) ^{n+1}\left( \dfrac{1}{n}\right)\) | \( \{ e_{n}\} =\left\{( -1) ^{n+1}\left( \dfrac{1}{n}\right) \right\}\) |