For each given geometric sequence, find the sum of the first: 
1. 5 terms of 4,12,36,108...

To find the sum of the first 5 terms of the geometric sequence 4, 12, 36, 108, ..., we will use the formula for the sum of the first \( n \) terms of a geometric sequence:

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

Where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a_1 \) is the first term of the sequence,
- \( r \) is the common ratio, and
- \( n \) is the number of terms.

### Step 1: Identify the values
- \( a_1 = 4 \) (the first term),
- \( r = \frac{12}{4} = 3 \) (the common ratio),
- \( n = 5 \) (the number of terms).

### Step 2: Apply the formula

\[
S_5 = \frac{4 \cdot (3^5 - 1)}{3 - 1}
\]

### Step 3: Calculate \( 3^5 \)

\[
3^5 = 243
\]

### Step 4: Substitute and calculate

\[
S_5 = \frac{4 \cdot (243 - 1)}{2} = \frac{4 \cdot 242}{2} = \frac{968}{2} = 484
\]

So, the sum of the first 5 terms is **484**.

Would you like any further details or have any questions?

Here are 5 related questions:
1. What is the sum of the first 6 terms of this sequence?
2. How do you find the 7th term of the sequence?
3. What is the sum of an infinite geometric series with the same first term and ratio?
4. How do you determine if a series converges or diverges?
5. How does the sum of the first \( n \) terms change if the common ratio is less than 1?

**Tip:** When calculating the sum of a geometric series, double-check the common ratio and the number of terms to ensure the accuracy of your result.

Learn how to find the sum of the first 5 terms of the geometric sequence 4, 12, 36, 108... using the sum of geometric series formula. Detailed step-by-step solution provided.

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