Find the sum $$ \sum_{n=1}^{12} 5 \cdot 5^{n-1}= $$

Question

UpStudy AI · Accepted Answer

To find the sum $$ \sum_{n=1}^{12} 5 \cdot 5^{n-1} $$, we can first recognize that this is a geometric series.

The general form of a geometric series is given by:

$$
S = a \cdot \frac{1 - r^n}{1 - r}
$$

where:
- $$ S $$ is the sum of the series,
- $$ a $$ is the first term,
- $$ r $$ is the common ratio,
- $$ n $$ is the number of terms.

In our case:
- The first term $$ a = 5 \cdot 5^{1-1} = 5 \cdot 5^0 = 5 $$.
- The common ratio $$ r = 5 $$.
- The number of terms $$ n = 12 $$.

Now we can substitute these values into the formula:

$$
S = 5 \cdot \frac{1 - 5^{12}}{1 - 5}
$$

Calculating the denominator:

$$
1 - 5 = -4
$$

Now substituting back into the sum formula:

$$
S = 5 \cdot \frac{1 - 5^{12}}{-4}
$$

This simplifies to:

$$
S = -\frac{5}{4} (1 - 5^{12})
$$

Now we need to calculate $$ 5^{12} $$. Let's compute that first.

Calculating $$ 5^{12} $$:

$$
5^{12} = 244140625
$$

Now substituting this value back into the sum:

$$
S = -\frac{5}{4} (1 - 244140625) = -\frac{5}{4} (-244140624) = \frac{5 \cdot 244140624}{4}
$$

Now we can calculate $$ \frac{5 \cdot 244140624}{4} $$:

$$
5 \cdot 244140624 = 1220703120
$$

Now dividing by 4:

$$
\frac{1220703120}{4} = 305175780
$$

Thus, the sum $$ \sum_{n=1}^{12} 5 \cdot 5^{n-1} $$ is:

$$
\boxed{305175780}
$$
The sum $$ \sum_{n=1}^{12} 5 \cdot 5^{n-1} $$ is 305,175,780.

Find the sum \( \sum_{n=1}^{12} 5 \cdot 5^{n-1}= \)