In a series, the first term is k. Each term thereafter is three times the preceding term. The sum of the first six terms is 728. What is the sum of the first three terms of the series ?

Options

A. 2
B. 13
C. 26
D. 80

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Define the terms of the series
The first term of the series is given as \( k \). Each subsequent term is three times the preceding term. Therefore, we can express the first six terms of the series as follows:
- First term: \( k \)
- Second term: \( 3k \)
- Third term: \( 3^2 k = 9k \)
- Fourth term: \( 3^3 k = 27k \)
- Fifth term: \( 3^4 k = 81k \)
- Sixth term: \( 3^5 k = 243k \)

### Step 2: Write the sum of the first six terms
The sum of the first six terms can be expressed as:
\[ S_6 = k + 3k + 9k + 27k + 81k + 243k \]
This can be simplified to:
\[ S_6 = k(1 + 3 + 9 + 27 + 81 + 243) \]

### Step 3: Calculate the sum of the coefficients
Now, we calculate the sum of the coefficients:
\[ 1 + 3 + 9 + 27 + 81 + 243 = 364 \]
Thus, we have:
\[ S_6 = k \cdot 364 \]

### Step 4: Set up the equation based on the given information
According to the problem, the sum of the first six terms is equal to 728. Therefore, we set up the equation:
\[ 364k = 728 \]

### Step 5: Solve for \( k \)
To find \( k \), we divide both sides of the equation by 364:
\[ k = \frac{728}{364} = 2 \]

### Step 6: Find the sum of the first three terms
Now that we have \( k = 2 \), we can find the sum of the first three terms:
- First term: \( k = 2 \)
- Second term: \( 3k = 3 \times 2 = 6 \)
- Third term: \( 9k = 9 \times 2 = 18 \)

The sum of the first three terms is:
\[ S_3 = k + 3k + 9k = 2 + 6 + 18 = 26 \]

### Conclusion
Thus, the sum of the first three terms of the series is \( \boxed{26} \).