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The first three terms of a series are given by:

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\$$\\var{a} + \\simplify{{a}*{r}} + \\simplify{{a}*{r}^2}\\,+ \\, ...........\$$

Find the nth term of a Geometric progression

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Calculate the \$$\\var{n}th\$$ term of the series.

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\$$T_\\var{n}=\$$ [[0]]

If the ratio between successive pairs of terms is a constant then the series under examination is a geometric progression.

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Ths first term is \$$a\$$ and the common ratio is \$$r\$$.

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The formula for the nth term of the series is given by:    \$$T_n=ar^{n-1}\$$

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In this example \$$a=\\var{a}\$$,   \$$r = \\frac{\\simplify{{a}*{r}}}{\\var{a}}=\\var{r}\$$  and  \$$n = \\var{n}\$$

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\$$T_\\var{n}=\\var{a}*\\var{r}^{\\simplify{{n}-1}}\$$

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\$$T_\\var{n}=\\var{a}*\\simplify{{r}^{{n}-1}}\$$

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\$$T_\\var{n}=\\simplify{{a}*{r}^{{n}-1}}\$$

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