A geometric series has a first term of a (a ≠ 0) and common ratio r (0 < r < 1). If the sum of the
first 8 term is equal to half of the infinite sum, find the value of r up to 3 decimal points. Also
given, the 17th term of the series is 10, find a.

Let's solve the problem step by step.

### Step 1: Express the given information
Given:
- First term: \( a \)
- Common ratio: \( r \) (where \( 0 < r < 1 \))
- Sum of the first 8 terms: \( S_8 \)
- The sum of the infinite series: \( S_{\infty} \)
- \( S_8 = \frac{1}{2} S_{\infty} \)
- The 17th term of the series is 10.

### Step 2: Formulate the sum of the first 8 terms and the infinite sum

For a geometric series, the sum of the first \( n \) terms is given by:

\[
S_n = a \frac{1 - r^n}{1 - r}
\]

The sum of an infinite geometric series is given by:

\[
S_{\infty} = \frac{a}{1 - r}
\]

### Step 3: Use the given relationship between \( S_8 \) and \( S_{\infty} \)

Given \( S_8 = \frac{1}{2} S_{\infty} \):

\[
a \frac{1 - r^8}{1 - r} = \frac{1}{2} \cdot \frac{a}{1 - r}
\]

We can cancel out \( \frac{a}{1 - r} \) (since \( a \neq 0 \) and \( r \neq 1 \)):

\[
1 - r^8 = \frac{1}{2}
\]

This simplifies to:

\[
r^8 = \frac{1}{2}
\]

Taking the 8th root of both sides:

\[
r = \left(\frac{1}{2}\right)^{\frac{1}{8}}
\]

Let's calculate \( r \) to three decimal places:

\[
r = 0.5^{\frac{1}{8}} \approx 0.917
\]

### Step 4: Find the value of \( a \)

The 17th term of a geometric series is given by:

\[
T_{17} = ar^{16}
\]

Given that \( T_{17} = 10 \), we have:

\[
10 = a \times 0.917^{16}
\]

Now, solve for \( a \):

\[
a = \frac{10}{0.917^{16}}
\]

Let's calculate this:

\[
a \approx \frac{10}{0.349} \approx 28.653
\]

### Final Answer:
- The value of \( r \) is approximately \( 0.917 \).
- The value of \( a \) is approximately \( 28.653 \).

Would you like a detailed breakdown of any part of this solution, or do you have any questions?

### Relative Questions:
1. What is the general formula for the sum of the first \( n \) terms in a geometric series?
2. How does the common ratio \( r \) affect the sum of an infinite geometric series?
3. Can you derive the sum of an infinite geometric series if \( r \) is not between 0 and 1?
4. What happens to the sum \( S_n \) as \( n \) approaches infinity when \( 0 < r < 1 \)?
5. How would the value of \( a \) change if \( r \) was greater than 1?
6. What is the relationship between the 8th term and the sum of the first 8 terms?
7. How does the sum of the first 8 terms compare with the infinite sum as \( r \) changes?
8. Can you find the sum of the first 17 terms with the given \( a \) and \( r \)?

### Tip:
When working with geometric series, always ensure that the common ratio \( r \) is in the appropriate range for the sum to be meaningful, especially for infinite sums.

Learn how to solve a geometric series problem involving the sum of the first 8 terms being half of the infinite sum, and the 17th term given as 10. Find the common ratio r up to three decimal points and determine the first term a. This problem involves key concepts in geometric series, exponential growth, and series summation. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation