Pregunta
Find the sum of the first 10 terms of the following geometric sequences: \( \{1.5,3,6,12,24 \ldots\} \) 1526.0 1532.0 1534.5 1528.5
Ask by Washington Washington. in the United States
Jan 26,2025
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The sum of the first 10 terms is 1534.5.
Solución
To find the sum of the first 10 terms of the geometric sequence \( \{1.5, 3, 6, 12, 24, \ldots\} \), follow these steps:
1. **Identify the first term and the common ratio:**
- First term (\( a \)) = 1.5
- Common ratio (\( r \)) = \( \frac{3}{1.5} = 2 \)
2. **Use the formula for the sum of the first \( n \) terms of a geometric sequence:**
\[
S_n = a \frac{r^n - 1}{r - 1}
\]
Plugging in the values:
\[
S_{10} = 1.5 \frac{2^{10} - 1}{2 - 1} = 1.5 (1024 - 1) = 1.5 \times 1023 = 1534.5
\]
**Answer:** 1534.5
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To find the sum of the first 10 terms of the geometric sequence \( \{1.5, 3, 6, 12, 24, \ldots\} \), we identify the first term \( a = 1.5 \) and the common ratio \( r = \frac{3}{1.5} = 2 \). The formula for the sum \( S_n \) of the first \( n \) terms of a geometric series is: \[ S_n = a \frac{1 - r^n}{1 - r} \] Plugging in the values for \( n = 10 \): \[ S_{10} = 1.5 \frac{1 - 2^{10}}{1 - 2} = 1.5 \frac{1 - 1024}{-1} = 1.5 \times 1023 = 1534.5 \] So, the sum of the first 10 terms is \( 1534.5 \).
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