Notes: `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)

Wednesday, June 3, 2015

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

$\displaystyle{a}_{{n}}=\frac{{{3}}^{{{n}-{1}}}}{{{\left({n}-{1}\right)}!}}$

We have to put $\displaystyle{n}-{1}$ because both powers and factorials start with 0 ( $\displaystyle{a}_{{1}}=\frac{{{3}}^{{0}}}{{{0}!}}$ )

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

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

$\displaystyle{a}_{{n}}=\frac{{{3}}^{{{n}-{1}}}}{{{\left({n}-{1}\right)}!}}$

We have to put $\displaystyle{n}-{1}$ because both powers and factorials start with 0 ( $\displaystyle{a}_{{1}}=\frac{{{3}}^{{0}}}{{{0}!}}$ )

Notes

Wednesday, June 3, 2015

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

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Is Charlotte Bronte's Jane Eyre a feminist novel?

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Wednesday, June 3, 2015

Write an expression for the apparent nth term of the sequence. (assume that n begins with 1)

No comments:

Post a Comment

Is Charlotte Bronte&#39;s Jane Eyre a feminist novel?

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

Is Charlotte Bronte's Jane Eyre a feminist novel?