https://file.aiquickdraw.com/mathbot/file/1727230350386dhk88kbz.jpg

The image contains two mathematical series problems written in Indonesian. Here's a breakdown and solution for each:

### 1. Series 1: \( S_{10} \)

The series is:
\[ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots \]

This is a geometric series where the first term \( a = 1 \) and the common ratio \( r = \frac{1}{2} \).

The sum of the first \( n \) terms of a geometric series is given by:
\[ S_n = a \frac{1 - r^n}{1 - r} \]

For \( n = 10 \), the sum \( S_{10} \) is:
\[
S_{10} = 1 \times \frac{1 - \left(\frac{1}{2}\right)^{10}}{1 - \frac{1}{2}} 
= \frac{1 - \frac{1}{1024}}{\frac{1}{2}} 
= 2 \times \left(1 - \frac{1}{1024}\right) 
= 2 \times \frac{1023}{1024}
= \frac{2046}{1024}
\approx 1.998
\]

So, \( S_{10} \approx 1.998 \).

### 2. Series 2: \( S_{15} \)

The series is:
\[ 5 + 15 + 45 + \dots \]

This is also a geometric series where the first term \( a = 5 \) and the common ratio \( r = 3 \).

The sum of the first \( n \) terms of a geometric series is given by:
\[ S_n = a \frac{r^n - 1}{r - 1} \]

For \( n = 15 \), the sum \( S_{15} \) is:
\[
S_{15} = 5 \times \frac{3^{15} - 1}{3 - 1} 
= \frac{5 \times (3^{15} - 1)}{2}
\]

Calculating \( 3^{15} \):
\[
3^{15} = 14348907
\]

Thus,
\[
S_{15} = \frac{5 \times (14348907 - 1)}{2} = \frac{5 \times 14348906}{2} = \frac{71744530}{2} = 35872265
\]

So, \( S_{15} = 35,872,265 \).

---

Would you like more details or explanations on any part of the solution? 

Here are some related questions:

1. How do you derive the formula for the sum of a geometric series?
2. What is the significance of the common ratio in determining the sum of a geometric series?
3. Can you explain why the series \( 1 + \frac{1}{2} + \frac{1}{4} + \dots \) converges?
4. How does the sum of an infinite geometric series differ from the sum of a finite geometric series?
5. What happens to the sum \( S_n \) as \( n \) approaches infinity in a geometric series where \( |r| < 1 \)?

**Tip:** Always check the common ratio in a series to determine whether the series converges (if \( |r| < 1 \)) or diverges (if \( |r| \geq 1 \)).

1. 1 + 1/2 + 1/4 + 1/8 + ... Tentukan S10. 2. 5 + 15 + 45 + ... Tentukan S15.

Learn how to find the sum of the geometric series 1 + 1/2 + 1/4 + ... for the first 10 terms (S10) and 5 + 15 + 45 + ... for the first 15 terms (S15). This guide covers geometric series formulas and step-by-step solutions, suitable for 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