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

Options

A. `27^(th)`
B. `28^(th)`
C. `29^(th)`
D. No such term exits

Answer

To find the first negative term of the sequence 20, 19 1/4, 18 1/2, 17 3/4, ..., we first need to determine if this sequence is an arithmetic progression (AP) and then find the general term.

### Step 1: Identify the sequence type
The given sequence is:
- First term (a) = 20
- Second term = 19 1/4 = 19.25
- Third term = 18 1/2 = 18.5
- Fourth term = 17 3/4 = 17.75

To check if it is an AP, we calculate the common difference (d) between the terms.

### Step 2: Calculate the common difference
1. **Difference between the first and second terms:**
\[ d = 19.25 - 20 = -0.75 \]

2. **Difference between the second and third terms:**
\[ d = 18.5 - 19.25 = -0.75 \]

3. **Difference between the third and fourth terms:**
\[ d = 17.75 - 18.5 = -0.75 \]

Since the difference is consistent, the sequence is an arithmetic progression with a common difference \( d = -0.75 \).

### Step 3: Write the formula for the nth term
The formula for the nth term of an arithmetic progression is given by:
\[ a_n = a + (n - 1) \cdot d \]
Substituting the values:
\[ a_n = 20 + (n - 1)(-0.75) \]
Simplifying this:
\[ a_n = 20 - 0.75(n - 1) \]
\[ a_n = 20 - 0.75n + 0.75 \]
\[ a_n = 20.75 - 0.75n \]

### Step 4: Find when the term becomes negative
To find the first negative term, we set the nth term less than 0:
\[ 20.75 - 0.75n < 0 \]
Solving for \( n \):
\[ 20.75 < 0.75n \]
\[ n > \frac{20.75}{0.75} \]
Calculating this:
\[ n > 27.6667 \]

### Step 5: Determine the smallest integer n
Since \( n \) must be an integer, we round up to the next whole number:
\[ n = 28 \]

### Conclusion
The first negative term of the sequence occurs at the 28th term.