To solve this problem, let's break it down step by step.

Step 1: Formula for the terms of a geometric sequence

The general formula for the $n$-th term of a geometric sequence is given by:

$a_n = a_1 \cdot r^{n-1}$

where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term,
• $r$ is the common ratio.

Step 2: Use known values to find the common ratio $r$

From the problem, we know:

• The first term $a_1 = 4$,
• The fourth term $a_4 = -2.916$.

Using the formula for the $n$-th term: $a_4 = a_1 \cdot r^{4-1} = a_1 \cdot r^3$ Substitute the values: $-2.916 = 4 \cdot r^3$ Solve for $r^3$: $r^3 = \frac{-2.916}{4} = -0.729$

Now, take the cube root of both sides to find $r$: $r = \sqrt[3]{-0.729} = -0.9$

Step 3: Find the sum to infinity

The formula for the sum to infinity of a geometric series is given by:

$S_\infty = \frac{a_1}{1 - r}$ This formula is valid when $|r| < 1$, which is true here as $r = -0.9$.

Substitute the values: $S_\infty = \frac{4}{1 - (-0.9)} = \frac{4}{1 + 0.9} = \frac{4}{1.9}$ Now, calculate: $S_\infty = 2.105$

Final Answer:

The sum to infinity of the given geometric sequence is approximately 2.105.

Would you like any details or clarifications?

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Here are five related questions to explore further:

1. What is the sum of the first 5 terms of this sequence?
2. How does the common ratio affect the convergence of a geometric series?
3. Can a geometric series with $r > 1$ converge? Why or why not?
4. What is the sum to infinity if the common ratio was $0.5$ instead of $-0.9$?
5. How do we find the sum of a finite geometric series?

Tip: The sum to infinity of a geometric series only converges if the absolute value of the common ratio is less than 1 ($|r| < 1$).