Find the sum of  12 terms of the series 48, 24, 12...

The given series is:

\[
48, 24, 12, \dots
\]

This is a **geometric series** where the first term \(a_1 = 48\) and the common ratio \(r = \frac{24}{48} = \frac{1}{2}\).

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

\[
S_n = a_1 \cdot \frac{1 - r^n}{1 - r}
\]

For this series, we need to find the sum of the first 12 terms (\(S_{12}\)):

\[
S_{12} = 48 \cdot \frac{1 - \left(\frac{1}{2}\right)^{12}}{1 - \frac{1}{2}}
\]

Let's compute this step by step:

1. \(r^{12} = \left(\frac{1}{2}\right)^{12} = \frac{1}{4096}\)
2. \(1 - \left(\frac{1}{2}\right)^{12} = 1 - \frac{1}{4096} = \frac{4096 - 1}{4096} = \frac{4095}{4096}\)
3. \(S_{12} = 48 \cdot \frac{\frac{4095}{4096}}{\frac{1}{2}} = 48 \cdot 2 \cdot \frac{4095}{4096} = 96 \cdot \frac{4095}{4096} = \frac{393120}{4096} \approx 95.76\)

So, the sum of the first 12 terms is approximately:

\[
\boxed{95.76}
\]

Would you like more details on geometric series or have any further questions?

Here are 5 related questions for practice:
1. What is the sum of the first 10 terms of the series \( 100, 50, 25, \dots \)?
2. How can you find the sum of an infinite geometric series?
3. What is the sum of the first 5 terms of the series \( 3, 6, 12, \dots \)?
4. How do you find the common ratio in a geometric series?
5. What is the sum of the first 15 terms of the series \( 80, 40, 20, \dots \)?

**Tip:** When calculating the sum of geometric series, be mindful of whether the common ratio is greater than or less than 1. It affects convergence for infinite series!

Find the sum of 12 terms of the series 48, 24, 12...

Learn how to find the sum of the first 12 terms of a geometric series with the first term 48 and common ratio 1/2. This problem covers key concepts in geometric series and the summation formula. Ideal for students in Grades 9-12.

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