the nth term of a geometric sequence is (2)(-3)^n-3. if the sum of the first k term of the sequence is greater than 2024, find the least value of k 

Given the geometric sequence with the nth term as \( a_n = 2(-3)^{n-3} \), we need to find the least value of \( k \) such that the sum of the first \( k \) terms exceeds 2024.

First, let's rewrite the nth term \( a_n \):
\[ a_n = 2(-3)^{n-3} = \frac{2(-3)^n}{(-3)^3} = \frac{2(-3)^n}{-27} = -\frac{2}{27} \cdot (-3)^n \]

Thus, the nth term is:
\[ a_n = -\frac{2}{27} \cdot (-3)^n \]

Next, we know that the sum of the first \( k \) terms of a geometric sequence is given by:
\[ S_k = a \frac{1-r^k}{1-r} \]

where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = -\frac{2}{27} \cdot (-3)^1 = \frac{6}{27} = \frac{2}{9} \) and \( r = -3 \).

Substituting these values into the sum formula:
\[ S_k = \frac{\frac{2}{9}(1 - (-3)^k)}{1 - (-3)} = \frac{\frac{2}{9}(1 - (-3)^k)}{4} = \frac{2}{36}(1 - (-3)^k) = \frac{1}{18}(1 - (-3)^k) \]

We need to find the smallest \( k \) such that \( S_k > 2024 \):
\[ \frac{1}{18}(1 - (-3)^k) > 2024 \]

Multiply both sides by 18:
\[ 1 - (-3)^k > 2024 \times 18 \]
\[ 1 - (-3)^k > 36432 \]

Subtract 1 from both sides:
\[ -(-3)^k > 36431 \]
\[ (-3)^k < -36431 \]

Since \((-3)^k\) is negative for odd \( k \), and positive for even \( k \), we look at odd \( k \) values:
\[ 3^k > 36431 \]

Now, find the smallest \( k \) such that this inequality holds. We can use logarithms:
\[ k \log 3 > \log 36431 \]
\[ k > \frac{\log 36431}{\log 3} \]

Using logarithm values:
\[ \log 36431 \approx 4.5615 \]
\[ \log 3 \approx 0.4771 \]

Thus:
\[ k > \frac{4.5615}{0.4771} \approx 9.56 \]

Since \( k \) must be an integer, and an odd number:
\[ k = 11 \]

Therefore, the least value of \( k \) such that the sum of the first \( k \) terms is greater than 2024 is \( k = 11 \).

Would you like more details or have any questions?

Here are 8 relative questions to expand your understanding:
1. What is the general formula for the nth term of a geometric sequence?
2. How do you derive the sum of the first \( k \) terms of a geometric sequence?
3. What are the characteristics of a geometric sequence with a common ratio greater than 1?
4. What happens to the sum of an infinite geometric series if the common ratio is between -1 and 1?
5. How do you handle geometric sequences with negative common ratios in summation formulas?
6. Can you explain the significance of the common ratio in determining the behavior of a geometric sequence?
7. How does the value of the common ratio affect the convergence or divergence of a geometric series?
8. What are real-life examples of geometric sequences and series?

**Tip:** When working with sequences and series, always verify the common ratio and initial terms to avoid calculation errors.

Learn how to find the least value of k such that the sum of the first k terms of a geometric sequence, defined by a specific nth term formula, exceeds 2024. Includes step-by-step solution using geometric sequence formulas and exponential functions.

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