Notes: `a_1 = 80, a_(k + 1) = (-1/2)a_k` Write the first five terms of the geometric sequence. Determine the common ratio and write the nth term of...

Friday, January 10, 2014

$\displaystyle{a}_{{1}}={80},{a}_{{{k}+{1}}}={\left(-\frac{{1}}{{2}}\right)}{a}_{{k}}$ Write the first five terms of the geometric sequence. Determine the common ratio and write the nth term of...

The given are:

$\displaystyle{a}_{{1}}={80}$

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

To determine the first five terms of the geometric sequence, plug-in the values k=1,2,3,4 to the given recursive formula.

When k=1, the nth term is:

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

$\displaystyle{a}_{{2}}={\left(-\frac{{1}}{{2}}\right)}\cdot{80}$

$\displaystyle{a}_{{2}}=-{40}$

When k=2, the nth term is:

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

$\displaystyle{a}_{{3}}={\left(-\frac{{1}}{{2}}\right)}\cdot{\left(-{40}\right)}$

$\displaystyle{a}_{{3}}={20}$

When k=3, the nth term is:

$\displaystyle{a}_{{{3}+{1}}}={\left(-\frac{{1}}{{2}}\right)}{a}_{{3}}$

$\displaystyle{a}_{{4}}={\left(-\frac{{1}}{{2}}\right)}\cdot{20}$

$\displaystyle{a}_{{4}}=-{10}$

And when k=4, the nth term is:

$\displaystyle{a}_{{{4}+{1}}}={\left(-\frac{{1}}{{2}}\right)}{a}_{{4}}$

$\displaystyle{a}_{{5}}={\left(-\frac{{1}}{{2}}\right)}\cdot{\left(-{10}\right)}$

$\displaystyle{a}_{{5}}={5}$

Therefore, the first five terms of the geometric sequence are $\displaystyle{\left\lbrace{80},-{40},\ldots\right.}$

The given are:

$\displaystyle{a}_{{1}}={80}$

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

To determine the first five terms of the geometric sequence, plug-in the values k=1,2,3,4 to the given recursive formula.

When k=1, the nth term is:

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

$\displaystyle{a}_{{2}}={\left(-\frac{{1}}{{2}}\right)}\cdot{80}$

$\displaystyle{a}_{{2}}=-{40}$

When k=2, the nth term is:

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

$\displaystyle{a}_{{3}}={\left(-\frac{{1}}{{2}}\right)}\cdot{\left(-{40}\right)}$

$\displaystyle{a}_{{3}}={20}$

When k=3, the nth term is:

$\displaystyle{a}_{{{3}+{1}}}={\left(-\frac{{1}}{{2}}\right)}{a}_{{3}}$

$\displaystyle{a}_{{4}}={\left(-\frac{{1}}{{2}}\right)}\cdot{20}$

$\displaystyle{a}_{{4}}=-{10}$

And when k=4, the nth term is:

$\displaystyle{a}_{{{4}+{1}}}={\left(-\frac{{1}}{{2}}\right)}{a}_{{4}}$

$\displaystyle{a}_{{5}}={\left(-\frac{{1}}{{2}}\right)}\cdot{\left(-{10}\right)}$

$\displaystyle{a}_{{5}}={5}$

Therefore, the first five terms of the geometric sequence are $\displaystyle{\left\lbrace{80},-{40},{20},-{10},{5}\right\rbrace}$ .

To determine the common ratio, apply the formula:

$\displaystyle{r}=\frac{{a}_{{{n}+{1}}}}{{a}_{{n}}}$

So the ratio of the consecutive terms of the geometric sequence is:

$\displaystyle{r}=\frac{{a}_{{5}}}{{a}_{{4}}}=\frac{{5}}{{-{10}}}=-\frac{{1}}{{2}}$

$\displaystyle{r}=\frac{{a}_{{4}}}{{a}_{{3}}}=\frac{{-{10}}}{{20}}=-\frac{{1}}{{2}}$

$\displaystyle{r}=\frac{{a}_{{3}}}{{a}_{{2}}}=\frac{{20}}{{-{40}}}=-\frac{{1}}{{2}}$

$\displaystyle{r}=\frac{{a}_{{2}}}{{a}_{{1}}}=\frac{{-{40}}}{{80}}=-\frac{{1}}{{2}}$

Thus, the common ratio of the geometric sequence is $\displaystyle-\frac{{1}}{{2}}$ .

To determine the nth term of geometric sequence, apply the formula:

$\displaystyle{a}_{{n}}={a}_{{1}}\cdot{{r}}^{{{n}-{1}}}$

Plugging in the values of a1 and r, the formula becomes:

$\displaystyle{a}_{{n}}={80}\cdot{{\left(-\frac{{1}}{{2}}\right)}}^{{{n}-{1}}}$

Hence, the nth term rule of this geometric sequence is $\displaystyle{a}_{{n}}={80}\cdot{{\left(-\frac{{1}}{{2}}\right)}}^{{{n}-{1}}}$ .

Notes

Friday, January 10, 2014

$\displaystyle{a}_{{1}}={80},{a}_{{{k}+{1}}}={\left(-\frac{{1}}{{2}}\right)}{a}_{{k}}$ Write the first five terms of the geometric sequence. Determine the common ratio and write the nth term of...

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Is Charlotte Bronte's Jane Eyre a feminist novel?

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Friday, January 10, 2014

Write the first five terms of the geometric sequence. Determine the common ratio and write the nth term of...

No comments:

Post a Comment

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

$\displaystyle{a}_{{1}}={80},{a}_{{{k}+{1}}}={\left(-\frac{{1}}{{2}}\right)}{a}_{{k}}$ Write the first five terms of the geometric sequence. Determine the common ratio and write the nth term of...

Is Charlotte Bronte's Jane Eyre a feminist novel?