Figure

Find the $n$ th term of each of the following sequences. Assume that the indicated patterns continue.

Solution

Sequence	$nth$ term
$(a) \, \, e, \dfrac{e^{2}}{2},\dfrac{e^{3}}{3},\ldots$	$a_{n}=\dfrac{e^{n}}{n}$
$(b) \, \, 1,\dfrac{1}{3},\,\dfrac{1}{9},\,\dfrac{1}{27},\ldots$	$b_{n}=\left( \dfrac{1}{3}\right) ^{n-1}$
$(c) \, \, 1,\,4,\,9,\,16,\,25,\,\ldots$	$c_{n}=n^{2}$
$(d) \, \, \dfrac{2}{2},\,\dfrac{4}{3}, \dfrac{6}{4}, \dfrac{8}{5},\,\ldots$	$d_{n}=\dfrac{2n}{n+1}$
$(e) \, \, 1,\,-\dfrac{1}{2},\,\dfrac{1}{3},\,-\dfrac{1}{4},\,\dfrac{1}{5},\,\ldots$	$e_{n}=\dfrac{(-1)^{n+1}}{n}$
$(f) \, \, 1,\,\dfrac{1}{2},\,1,\,\dfrac{1}{4},\,1,\,\dfrac{1}{6},\ldots$	$f_{n}=\left\{\begin{array}{lll} 1 &\hbox{if } n\hbox{ is odd} \\ \dfrac{1}{n} &\hbox{if } n\hbox{ is even} \end{array}\right.$