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

To find the sum of the first six terms of the geometric series \(36 + 18 + 9 + \ldots\), we start by identifying the terms and the common ratio.

### Step 1: Identify the first term and common ratio

1. The first term (\(a\)) is 36.
2. 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

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

\[
S_n = a \frac{1 - r^n}{1 - r},
\]

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.

For this problem, \( a = 36 \), \( r = \frac{1}{2} \), and \( n = 6 \).

### Step 3: Plug in the values

\[
S_6 = 36 \cdot \frac{1 - \left(\frac{1}{2}\right)^6}{1 - \frac{1}{2}}.
\]

Simplify the expression:

1. Calculate \( \left(\frac{1}{2}\right)^6 \):

   \[
   \left(\frac{1}{2}\right)^6 = \frac{1}{64}.
   \]

2. Substitute back into the sum formula:

   \[
   S_6 = 36 \cdot \frac{1 - \frac{1}{64}}{\frac{1}{2}}.
   \]

3. Simplify the numerator:

   \[
   1 - \frac{1}{64} = \frac{64}{64} - \frac{1}{64} = \frac{63}{64}.
   \]

4. Simplify the fraction:

   \[
   S_6 = 36 \cdot \frac{\frac{63}{64}}{\frac{1}{2}} = 36 \cdot \frac{63}{64} \cdot 2.
   \]

5. Multiply:

   \[
   S_6 = 36 \cdot \frac{126}{64} = 36 \cdot \frac{63}{32} = \frac{36 \cdot 63}{32}.
   \]

6. Calculate the product:

   \[
   S_6 = \frac{2268}{32} = 70.875.
   \]

### Final Answer

\[
\boxed{70.875}
\]

Would you like further details or have any questions?

### Related Questions
1. What is the sum of the first eight terms of the series?
2. How does the sum change if the common ratio is altered?
3. Can you find the sum of an infinite series for this geometric sequence?
4. How do you determine if a geometric series converges?
5. What happens to the sum if the first term is doubled?

### Tip
For geometric series, the sum can be quickly found when \( |r| < 1 \) using the formula above, as it converges smoothly.

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

Explore how to find the sum of the first six terms of the geometric series 36 + 18 + 9 +... using the geometric series sum formula. This step-by-step guide covers key concepts in sequences and series. Suitable for high school students in Grades 9-12.

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