Geometric Series Summation: Find S10 and S15

Math Problem Statement

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

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

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:

= \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$$).

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Geometric Series

Sum of geometric series S_n = a * (1 - r^n) / (1 - r) when |r| < 1
Sum of geometric series S_n = a * (r^n - 1) / (r - 1) when r > 1

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