This problem involves solving for the number of terms in a geometric sequence. Let's solve it step by step:

Given:

1. The first term ($a$) is 4.
2. The sum of all terms ($S_n$) is $-59048$.
3. The common ratio ($r$) is $-3$.
4. The formula for the sum of a geometric sequence: $S_n = a \frac{1 - r^n}{1 - r}, \quad r \neq 1.$

Step 1: Substitute known values

Using the formula for the sum of the sequence: $-59048 = 4 \cdot \frac{1 - (-3)^n}{1 - (-3)}.$

Simplify the denominator: $-59048 = 4 \cdot \frac{1 - (-3)^n}{4}.$

Cancel the 4: $-59048 = 1 - (-3)^n.$

Rearrange to isolate $(-3)^n$: $(-3)^n = 1 + 59048.$

$(-3)^n = 59049.$

Step 2: Solve for $n$

Take the logarithm of both sides or solve by inspection: $(-3)^n = 3^{10}.$

Since the base is $-3$, $n$ must be 10 because the powers of $(-3)$ alternate signs. Here, the result matches the positive side.

Final Answer:

$n = 10.$

Let me know if you’d like further clarification or verification!

Related Questions:

1. How would the solution change if the common ratio were positive?
2. Can this method be generalized for other common ratios?
3. What happens if the sum $S_n$ is positive but $r$ is negative?
4. How would the formula change for an infinite geometric sequence?
5. What practical applications use geometric sequences with negative ratios?

Tip:

When solving equations involving powers, logarithms can simplify the process—especially when $n$ isn't an integer.