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

Answer

To find the first negative term of the sequence 20, 19(1/4), 18(1/2), 17(3/4), we will analyze the sequence step by step.

### Step 1: Identify the sequence and its pattern
The given sequence is:
- 20
- 19(1/4) = 19.25
- 18(1/2) = 18.5
- 17(3/4) = 17.75

### Step 2: Determine the common difference
To check if this sequence is an arithmetic progression (AP), we need to find the common difference (d) between consecutive terms.

Calculating the differences:
- From 20 to 19(1/4):
\( 19.25 - 20 = -0.75 \) or \( -\frac{3}{4} \)
- From 19(1/4) to 18(1/2):
\( 18.5 - 19.25 = -0.75 \) or \( -\frac{3}{4} \)
- From 18(1/2) to 17(3/4):
\( 17.75 - 18.5 = -0.75 \) or \( -\frac{3}{4} \)

Thus, the common difference \( d = -\frac{3}{4} \).

### Step 3: Write the general term of the AP
The nth term of an arithmetic progression can be expressed as:
\[ A_n = A + (n-1) \times d \]
where \( A \) is the first term and \( d \) is the common difference.

Here, \( A = 20 \) and \( d = -\frac{3}{4} \).
So, the nth term is:
\[ A_n = 20 + (n-1) \times \left(-\frac{3}{4}\right) \]

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

### Step 5: Solve the inequality
Rearranging the inequality:
\[ 20 - \frac{3}{4}(n-1) < 0 \]

Multiply through by 4 to eliminate the fraction:
\[ 80 - 3(n-1) < 0 \]
\[ 80 - 3n + 3 < 0 \]
\[ 83 - 3n < 0 \]
\[ 3n > 83 \]
\[ n > \frac{83}{3} \]
\[ n > 27.67 \]

### Step 6: Find the smallest integer n
The smallest integer greater than 27.67 is \( n = 28 \).

### Step 7: Calculate the 28th term
Now, we substitute \( n = 28 \) into the formula for \( A_n \):
\[ A_{28} = 20 + (28-1) \left(-\frac{3}{4}\right) \]
\[ A_{28} = 20 + 27 \left(-\frac{3}{4}\right) \]
\[ A_{28} = 20 - \frac{81}{4} \]
\[ A_{28} = 20 - 20.25 \]
\[ A_{28} = -0.25 \]

### Conclusion
The first negative term of the sequence is \( -0.25 \).

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