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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}\\,+ \\, ...........\$$

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Find the sum of the first n terms of a Geometric progression

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Calculate the sum of the first \$$\\var{n}\$$ terms of the series.

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\$$S_\\var{n}=\$$ [[0]]

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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 sum of the first \$$n\$$ terms of the series is given by:    \$$S_n=\\frac{a(1-r^{n})}{1-r}\$$

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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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\$$S_\\var{n}=\\frac{\\var{a}(1-(\\var{r})^{\\var{n}})}{1-\\var{r}}\$$

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\$$S_\\var{n}=\\frac{\\var{a}*(\\simplify{1-{r}^{n}})}{\\simplify{1-{r}}}\$$

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\$$S_\\var{n}=\\frac{\\simplify{{a}*(1-{r}^{n})}}{\\simplify{1-{r}}}\$$

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\$$S_\\var{n}=\\var{s}\$$

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