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The first term

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The index of a term to calculate.

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The range is picked so that the number is between 1,000 and 1,000,000.

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The common ratio

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Find the common ratio of a given geometric sequence, write down the formula for the nth term and use it to calculate a given term in the sequence.

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Find the common ratio for the following geometric series.

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

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Common ratio: [[0]]

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

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\$a_n = 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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Write down the formula for the  $n^\\text{th}$ term in the sequence

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$a_n =$ [[0]]

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What is the $\\var{n}^\\text{th}$ term in this sequence?

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$a_\\var{n} =$ [[0]]

"}], "tags": ["common ratio", "geometric sequences", "nth term", "sequences", "taxonomy"], "extensions": [], "functions": {}, "statement": "", "name": "Jo\u00ebl's copy of Find and use the formula for a geometric sequence", "ungrouped_variables": ["a", "r", "n", "nth_term"], "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
 $n$ $1$ $2$ $3$ $4$ $a_n$ $\\var{a}$ $\\var{a*r}$ $\\var{a*r^2}$ $\\var{a*r^3}$ $a_n \\div a_{n-1}$ $\\var{r}$ $\\var{r}$ $\\var{r}$
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The common ratio is $\\var{r}$.

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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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\$a_n = \\simplify[]{ {a}*{r}^n } \\text{.} \$

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#### c)

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We know from part b) that the formula for the $n^\\text{th}$ term is $a_n = \\simplify[]{ {a}*{r}^n}$.

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

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\\begin{align}
a_\\var{n} &= \\var{a} \\times \\var{r}^{\\var{b}} \\\\
&= \\var{a*r^n}
\\end{align}

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