// Numbas version: exam_results_page_options {"name": "Geometric Sequence - negative ratio ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "

The terms in a geometric sequence are found by repeatedly multiplying the last term by a constant, called the common ratio.

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a)

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To find the common ratio, pick a term of the sequence and divide it by the previous term.

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We can calculate the common ratio using a table:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$n$$1$$2$$3$$4$
$a_n$$\\var{a*r}$$\\var{a*r^2}$$\\var{a*r^3}$$\\var{a*r^4}$
Common ratio$\\displaystyle\\frac{\\var{a*r^2}}{(\\var{a*r})} = \\var{r}$$\\displaystyle\\frac{\\var{a*r^3}}{\\var{a*r^2}} = \\var{r}$$\\displaystyle\\frac{\\var{a*r^4}}{(\\var{a*r^3})} = \\var{r}$
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The common ratio is $\\var{d}$.

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b)

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The general formula for the $n^\\text{th}$ term of a geometric sequence is

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\\[\\displaystyle {a_n=ar^{(n-1)}\\text{,}}\\]

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where $a$ is the first term, and $r$ is the common ratio.

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So the formula for this sequence is

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\\[
\\begin{align}
a_n&=ar^{(n-1)}\\\\
&=\\var{a*r}\\times(\\var{r})^{(n-1)}\\\\
&=(\\var{a} \\times (\\var{r}))(\\var{r})^{n-1}\\\\
&=\\var{a}(\\var{r})^n\\text{.}
\\end{align}
\\]

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c)

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We know from part b) that the $n^{th}$ term for this sequence is $a_n = \\var{a}(\\var{r})^n$.

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Therefore the $\\var{nth}^{th}$ term in the sequence is 

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\\[
\\begin{align}
a_\\var{nth} &= \\var{a}(\\var{r})^\\var{nth}\\\\
&= \\var{a} \\times (\\var{{r}^{nth}})\\\\
&= \\var{{a}*{r}^{nth}}.
\\end{align}
\\]

\n

\n

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Find the common ratio for the following geometric series.

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$\\var{a*r}, \\var{a*r^2}, \\var{a*r^3}, \\var{a*r^4}\\ldots$

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Common ratio = [[0]]

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The formula for the $n^{th}$ term of a geometric sequence is

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\\[\\displaystyle{ar^{(n-1)}}\\]

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where $a$ is the first term in the sequence and $r$ is the common ratio.

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Find the formula for the  $n^{th}$ term in the sequence:

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$n^{th}$ term = [[0]]

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Find the value of the $\\var{nth}^{th}$ term.

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$a_\\var{nth}$ = [[0]]

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Given a geometric sequence, find the common ratio (negative in this question), write down the formula for the nth term and use it to calculate a given term.

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