Which term of the progression 19, `18(1)/(5), 17 (2)/(5)` ,..... is the first negative term?

Answer

To find the first negative term in the progression 19, \( 18 \frac{1}{5} \), \( 17 \frac{2}{5} \), ..., we can follow these steps:

### Step 1: Identify the first term and the common difference
The first term \( a \) of the progression is:
\[ a = 19 \]

To find the common difference \( d \), we calculate:
\[ d = 18 \frac{1}{5} - 19 \]
Convert \( 18 \frac{1}{5} \) to an improper fraction:
\[ 18 \frac{1}{5} = \frac{18 \times 5 + 1}{5} = \frac{90 + 1}{5} = \frac{91}{5} \]

Now, calculate \( d \):
\[ d = \frac{91}{5} - \frac{95}{5} = \frac{91 - 95}{5} = \frac{-4}{5} \]

### Step 2: Write the formula for the nth term
The nth term \( a_n \) of an arithmetic progression can be expressed as:
\[ a_n = a + (n - 1) \cdot d \]

Substituting the values of \( a \) and \( d \):
\[ a_n = 19 + (n - 1) \cdot \left(-\frac{4}{5}\right) \]

### Step 3: Set the nth term to be less than zero
To find the first negative term, we set:
\[ 19 + (n - 1) \cdot \left(-\frac{4}{5}\right) < 0 \]

### Step 4: Solve the inequality
Rearranging the inequality:
\[ (n - 1) \cdot \left(-\frac{4}{5}\right) < -19 \]

Multiplying both sides by \(-5/4\) (note that this reverses the inequality):
\[ n - 1 > \frac{95}{4} \]

Adding 1 to both sides:
\[ n > \frac{95}{4} + 1 = \frac{95}{4} + \frac{4}{4} = \frac{99}{4} \]

Calculating \( \frac{99}{4} \):
\[ \frac{99}{4} = 24.75 \]

Since \( n \) must be a whole number, the smallest integer greater than \( 24.75 \) is \( 25 \).

### Step 5: Conclusion
Thus, the first negative term occurs at:
\[ n = 25 \]