Which term of the sequence `20,19 1/4, 18 1/2, 17 3/4,...` is the first negative term?

Options

A. 27
B. 28
C. 29
D. None of these

Explanation

To find which term of the sequence \(20, 19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots\) is the first negative term, we can follow these steps:

### Step 1: Identify the first term (A) and the common difference (D)
The first term \(A\) is clearly \(20\).
To find the common difference \(D\), we can subtract the second term from the first term:
\[ D = 19 \frac{1}{4} - 20 = -\frac{3}{4} \]

### Step 2: Write the general formula for the nth term of the arithmetic progression (AP)
The nth term \(A_n\) of an AP can be expressed as:
\[ A_n = A + (n - 1)D \]
Substituting the values of \(A\) and \(D\):
\[ A_n = 20 + (n - 1)\left(-\frac{3}{4}\right) \]

### Step 3: Set up the inequality for the first negative term
We want to find the smallest \(n\) such that \(A_n < 0\):
\[ 20 + (n - 1)\left(-\frac{3}{4}\right) < 0 \]

### Step 4: Simplify the inequality
Rearranging the inequality gives:
\[ 20 - \frac{3}{4}(n - 1) < 0 \]
Multiplying through by \(4\) to eliminate the fraction:
\[ 80 - 3(n - 1) < 0 \]
Distributing the \(-3\):
\[ 80 - 3n + 3 < 0 \]
Combining like terms:
\[ 83 < 3n \]

### Step 5: Solve for \(n\)
Dividing both sides by \(3\):
\[ n > \frac{83}{3} \]
Calculating the division gives:
\[ n > 27.67 \]

### Step 6: Determine the smallest integer value of \(n\)
Since \(n\) must be a whole number, the smallest integer greater than \(27.67\) is \(28\).

### Conclusion
Thus, the first negative term of the sequence occurs at \(n = 28\).