In sequence S, the 3rd term is 4, the 2nd term is three times the 1st, and the 3rd term is four times the 2nd.What is the 1 st term is sequence S?

Options

A. 0
B. `1//3`
C. `1`
D. `3//2`

Answer

To solve the problem, we will define the terms of the sequence and then set up equations based on the information given in the question.

### Step 1: Define the terms of the sequence
Let:
- \( a_1 \) = the first term
- \( a_2 \) = the second term
- \( a_3 \) = the third term

### Step 2: Write down the information given in the question
According to the problem:
1. The third term \( a_3 \) is 4.
\[ a_3 = 4 \]
2. The second term \( a_2 \) is three times the first term \( a_1 \).
\[ a_2 = 3a_1 \]
3. The third term \( a_3 \) is four times the second term \( a_2 \).
\[ a_3 = 4a_2 \]

### Step 3: Substitute the value of \( a_3 \) into the third equation
From the first equation, we know \( a_3 = 4 \). Substitute this into the equation \( a_3 = 4a_2 \):
\[ 4 = 4a_2 \]

### Step 4: Solve for \( a_2 \)
To find \( a_2 \), divide both sides of the equation by 4:
\[ a_2 = 1 \]

### Step 5: Substitute \( a_2 \) back into the equation for \( a_1 \)
Now that we have \( a_2 = 1 \), we can substitute this value into the equation \( a_2 = 3a_1 \):
\[ 1 = 3a_1 \]

### Step 6: Solve for \( a_1 \)
To find \( a_1 \), divide both sides of the equation by 3:
\[ a_1 = \frac{1}{3} \]

### Conclusion
The first term \( a_1 \) in sequence S is:
\[ \boxed{\frac{1}{3}} \]