# How do you sum the series #1+a+a^2+a^3+...+a^n#?

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The solution George C.gives is very elegant. Here's another that has its own uses.

Let

Now subtract:

So, again:

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To sum the series (1 + a + a^2 + a^3 + \ldots + a^n), you can use the formula for the sum of a geometric series. The formula is:

[S = \frac{{a^{n+1} - 1}}{{a - 1}}]

Where:

- (S) is the sum of the series.
- (a) is the common ratio of the geometric series.
- (n) is the number of terms in the series.

So, to find the sum of the given series, plug in the values of (a) and (n) into the formula and calculate accordingly.

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