consider the geometric sequence with a first term of 4 and a fourth term of -2.916, find the sum to infinity of this sequence

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?

---

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 (\(|r| < 1\)).

Consider the geometric sequence with a first term of 4 and a fourth term of -2.916, find the sum to infinity of this sequence.

Learn how to find the sum to infinity of a geometric sequence with a first term of 4 and a fourth term of -2.916. This problem explores geometric sequences, the formula for the n-th term, and the conditions for series convergence. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation