The sum of the first 5 terms of the geometric progression   is  

What is the value of k?

To find the value of \( k \) for which the sum of the first 5 terms of a geometric progression equals \( k \), we'll use the formula for the sum of the first \( n \) terms of a geometric progression:

\[
S_n = \frac{a_1 (r^n - 1)}{r - 1}
\]

Here, \( S_5 \) (the sum of the first 5 terms) is given by:

\[
S_5 = \frac{a_1 (r^5 - 1)}{r - 1}
\]

Since you need the value of \( k \), the sum \( S_5 \) equals \( k \):

\[
k = \frac{a_1 (r^5 - 1)}{r - 1}
\]

To solve for \( k \), you need the values of the first term \( a_1 \) and the common ratio \( r \). From the previous problem, if we assume the same progression:

- The first term \( a_1 \) can be determined from \( a_1 \cdot r = 6 \).
- We already found that the common ratio \( r \) is \(-3\).

Let's calculate \( a_1 \):

\[
a_1 = \frac{6}{r} = \frac{6}{-3} = -2
\]

Now, substitute \( a_1 = -2 \) and \( r = -3 \) into the sum formula:

\[
k = \frac{-2 \cdot ((-3)^5 - 1)}{-3 - 1}
\]

Calculate \( (-3)^5 \):

\[
(-3)^5 = -243
\]

Now, substitute:

\[
k = \frac{-2 \cdot (-243 - 1)}{-4} = \frac{-2 \cdot (-244)}{-4} = \frac{488}{-4} = -122
\]

So, the value of \( k \) is \(-122\).

Would you like more details or have any other questions?

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Here are 5 related questions:

1. How would you find the sum of the first 10 terms of the geometric progression?
2. What is the sum of an infinite geometric series with the common ratio \(-3\)?
3. How does changing the common ratio affect the sum of the first \( n \) terms?
4. What is the sum of the first 4 terms of this progression?
5. How would the sum formula change if the common ratio were positive?

**Tip:** When dealing with geometric progressions, always double-check the sign of the common ratio as it directly impacts the terms' sign pattern and the sum calculation.

Learn how to calculate the sum of the first 5 terms of a geometric progression using the formula for the sum of a geometric series. Understand the concepts of first term, common ratio, and their impact on the series sum. Suitable for high school students in Grades 10-12.

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