Blog: `1, 3, 3^2/2, 3^3/6, 3^4/24, 3^5/120` Write an expression for the apparent nth term of the sequence. (assume that n begins with 1)

Saturday, February 20, 2016

$\displaystyle{1},{3},\frac{{{3}}^{{2}}}{{2}},\frac{{{3}}^{{3}}}{{6}},\frac{{{3}}^{{4}}}{{24}},\frac{{{3}}^{{5}}}{{120}}$ Write an expression for the apparent nth term of the sequence. (assume that n begins with 1)

Let us first write the first two terms as fractions as well.

$\displaystyle\frac{{1}}{{1}},\frac{{3}}{{1}},\frac{{{3}}^{{2}}}{{2}},\frac{{{3}}^{{3}}}{{6}},\frac{{{3}}^{{4}}}{{24}},\frac{{{3}}^{{5}}}{{120}},\ldots$

In the numerator we have powers of 3 ( $\displaystyle{{3}}^{{0}}={1}$ and $\displaystyle{{3}}^{{1}}={3}$ ). In the denominator we have factorials ( $\displaystyle{0}!={1},{1}!={1},{2}!={2},{3}!={6},{4}!={24},{5}!={120}$ ). Therefore, the $\displaystyle{n}$ th term of the sequence is