To find the sum of the given geometric series, we will first identify the components of the series and apply the formula for the sum of the first $n$ terms of a geometric series.

Step 1: General Formula for the Sum of a Geometric Series

The sum of the first $n$ terms of a geometric series is given by:

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

where:

• $a$ is the first term of the series,
• $r$ is the common ratio,
• $n$ is the number of terms to be summed.

Step 2: Identify the components of the series

The given series is:

$\sum_{k=1}^{6} -0.6 \cdot (-5)^{k-1}$

From this, we can see:

• The first term $a = -0.6 \cdot (-5)^{0} = -0.6 \cdot 1 = -0.6$,
• The common ratio $r = -5$,
• The number of terms $n = 6$.

Step 3: Apply the sum formula

Now, we substitute these values into the sum formula for a geometric series:

$S_6 = \frac{-0.6(1 - (-5)^6)}{1 - (-5)}$

Step 4: Simplify the expression

• First, calculate $(-5)^6 = 15625$,
• Then $1 - (-5) = 1 + 5 = 6$.

So, the sum becomes:

$S_6 = \frac{-0.6(1 - 15625)}{6} = \frac{-0.6(-15624)}{6}$

Now, simplify the expression:

$S_6 = \frac{9374.4}{6} = 1562.4$

Final Answer:

The sum of the first 6 terms of the series is $1562.4$.

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Would you like further explanation or a breakdown of the steps? Here are some related questions:

1. How does the formula for the sum of a geometric series change if the common ratio $r$ is greater than 1?
2. What happens if the common ratio $r$ is negative? How does this affect the behavior of the series?
3. How can we find the sum of an infinite geometric series?
4. What is the relationship between the sum of a geometric series and the convergence of the series?
5. How do you handle sums of geometric series when the number of terms $n$ is very large or infinite?

Tip: Always carefully identify the first term $a$ and common ratio $r$ when working with geometric series; these are the critical elements for applying the sum formula correctly.