Find the sum of the first sic term of the geometric series 36 + 18 + 9 + ...

To find the sum of the first six terms of the geometric series \(36 + 18 + 9 + \dots\), we can follow these steps:

### Step 1: Identify the first term (\(a\)) and the common ratio (\(r\)) of the geometric series.
- First term \(a = 36\).
- The common ratio \(r\) is found by dividing the second term by the first term:  
  \[
  r = \frac{18}{36} = \frac{1}{2}
  \]

### Step 2: Use the formula for the sum of the first \(n\) terms of a geometric series:
\[
S_n = \frac{a(1 - r^n)}{1 - r}
\]
where:
- \(S_n\) is the sum of the first \(n\) terms.
- \(a\) is the first term.
- \(r\) is the common ratio.
- \(n\) is the number of terms.

### Step 3: Substitute the known values into the formula.
For \(n = 6\), \(a = 36\), and \(r = \frac{1}{2}\):
\[
S_6 = \frac{36(1 - \left(\frac{1}{2}\right)^6)}{1 - \frac{1}{2}} = \frac{36(1 - \frac{1}{64})}{\frac{1}{2}}
\]

### Step 4: Simplify the expression.
\[
S_6 = \frac{36\left(\frac{63}{64}\right)}{\frac{1}{2}} = 36 \times \frac{63}{32} = \frac{2268}{32} = 70.875
\]

### Final Answer:
The sum of the first six terms of the geometric series is \(70.875\).

Would you like more details or have any questions?

Here are 5 related questions to deepen understanding:
1. What happens to the sum as the number of terms increases in a geometric series with \(r < 1\)?
2. How would the sum change if the common ratio were negative?
3. What is the sum of an infinite geometric series if \(r = \frac{1}{2}\)?
4. How do you derive the formula for the sum of a geometric series?
5. What is the sum of the first 10 terms of this series?

**Tip:** In a geometric series, if the common ratio's absolute value is less than 1, the series converges as the number of terms increases.

Find the sum of the first six terms of the geometric series 36 + 18 + 9 + ...

Learn how to find the sum of the first six terms of the geometric series 36 + 18 + 9 + ... with a step-by-step solution. This problem explores key concepts in sequences and series, including geometric series and the sum formula. Suitable for students in Grades 9-12.

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