Which term of the sequence, `4, 3 5/7, 3 3/7` …………. Is the first negative term?

Answer

To find the first negative term of the sequence \(4, 3 \frac{5}{7}, 3 \frac{3}{7}, \ldots\), we can follow these steps:

### Step 1: Identify the first term and the common difference
The first term \(a\) of the sequence is:
\[ a = 4 \]

Next, we need to find the common difference \(d\). We can calculate it using the first two terms:
\[ d = 3 \frac{5}{7} - 4 = \frac{26}{7} - \frac{28}{7} = -\frac{2}{7} \]

### Step 2: Write the general term of the arithmetic sequence
The \(n\)-th term \(T_n\) of an arithmetic sequence can be expressed as:
\[ T_n = a + (n - 1) \cdot d \]
Substituting the values of \(a\) and \(d\):
\[ T_n = 4 + (n - 1) \cdot \left(-\frac{2}{7}\right) \]

### Step 3: Set up the inequality for the first negative term
We want to find the smallest \(n\) such that \(T_n < 0\):
\[ 4 + (n - 1) \cdot \left(-\frac{2}{7}\right) < 0 \]

### Step 4: Solve the inequality
Rearranging the inequality:
\[ 4 - \frac{2(n - 1)}{7} < 0 \]
Multiplying through by 7 to eliminate the fraction:
\[ 28 - 2(n - 1) < 0 \]
Expanding and simplifying:
\[ 28 - 2n + 2 < 0 \implies 30 < 2n \implies n > 15 \]

### Step 5: Determine the smallest integer \(n\)
Since \(n\) must be an integer, the smallest integer greater than 15 is:
\[ n = 16 \]

### Step 6: Verify the result
To confirm, we can calculate \(T_{16}\):
\[ T_{16} = 4 + (16 - 1) \cdot \left(-\frac{2}{7}\right) = 4 - \frac{30}{7} = \frac{28}{7} - \frac{30}{7} = -\frac{2}{7} \]
Since \(-\frac{2}{7}\) is indeed negative, \(T_{16}\) is the first negative term.

### Conclusion
The first negative term of the sequence occurs at:
\[ \text{n = 16} \]