Given the series $$ \sum_{n=0}^{\infty} \frac{1}{(-5)^{2 n+1}} $$ answer the following questions. Answer Attempt 1 out of 2 $$ \sum_{n=0}^{\infty} \frac{1}{(-5)^{2 n+1}} \underbrace{\sum_{n}}_{n=0} \frac{1}{(-5)^{2 n+1}} $$ a geometric-series with the common ratio $$ r=\square $$ if and only if

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We start with the series

$$
\sum_{n=0}^{\infty} \frac{1}{(-5)^{2n+1}}.
$$

Notice that the term in the denominator can be rewritten as

$$
(-5)^{2n+1} = (-5)^{2n} \cdot (-5) = \left((-5)^2\right)^n \cdot (-5) = 25^n \cdot (-5).
$$

Thus, the general term becomes

$$
\frac{1}{(-5)^{2n+1}} = \frac{1}{25^n \cdot (-5)} = -\frac{1}{5} \cdot \frac{1}{25^n}.
$$

This allows us to express the series as

$$
-\frac{1}{5} \sum_{n=0}^{\infty} \frac{1}{25^n}.
$$

The series inside the summation

$$
\sum_{n=0}^{\infty} \frac{1}{25^n}
$$

is a geometric series with first term $$ a = 1 $$ and common ratio

$$
r = \frac{1}{25}.
$$

Thus, the original series is a geometric series (apart from the constant factor $$-\frac{1}{5}$$) if and only if the common ratio is

$$
r = \frac{1}{25}.
$$
The series is a geometric series with a common ratio of $$ \frac{1}{25} $$.

Given the series answer the following questions.
Answer Attempt 1 out of 2
a geometric-series with the common
ratio if and only if

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Given the series answer the following questions. Answer Attempt 1 out of 2 a geometric-series with the common ratio if and only if