Notes: `1, 3, 1, 3, 1, 3, 1` Write an expression for the apparent nth term of the sequence. (assume that n begins with 1)

Monday, August 17, 2015

$\displaystyle{1},{3},{1},{3},{1},{3},{1}$ Write an expression for the apparent nth term of the sequence. (assume that n begins with 1)

Let us start with the sequence $\displaystyle-{1},{1},-{1},{1},-{1},{1},\ldots$ The $\displaystyle{n}$ th term of this sequence is $\displaystyle{{\left(-{1}\right)}}^{{n}}$ (for odd $\displaystyle{n}$ we get $\displaystyle-{1}$ while for even $\displaystyle{n}$ we get $\displaystyle{1}$ ) If we add $\displaystyle{2}$ to each term of this sequence we get

$\displaystyle{2}-{1},{2}+{1},{2}-{1},{2}+{1},\ldots={1},{3},{1},{3},\ldots$

which is exactly our sequence. Therefore, the $\displaystyle{n}$ th term of our sequence is

$\displaystyle{a}_{{n}}={{\left(-{1}\right)}}^{{n}}+{2}$

...

Let us start with the sequence $\displaystyle-{1},{1},-{1},{1},-{1},{1},\ldots$ The $\displaystyle{n}$ th term of this sequence is $\displaystyle{{\left(-{1}\right)}}^{{n}}$ (for odd $\displaystyle{n}$ we get $\displaystyle-{1}$ while for even $\displaystyle{n}$ we get $\displaystyle{1}$ ) If we add $\displaystyle{2}$ to each term of this sequence we get

$\displaystyle{2}-{1},{2}+{1},{2}-{1},{2}+{1},\ldots={1},{3},{1},{3},\ldots$

which is exactly our sequence. Therefore, the $\displaystyle{n}$ th term of our sequence is

$\displaystyle{a}_{{n}}={{\left(-{1}\right)}}^{{n}}+{2}$

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