Which term of the sequence 2005,2000,1995,1990,1985,……. Is the first negative term

Answer

To find the first negative term of the sequence 2005, 2000, 1995, 1990, 1985, we can follow these steps:

### Step 1: Identify the Sequence Type
The given sequence is an arithmetic progression (AP) because the difference between consecutive terms is constant.

### Step 2: Find the First Term and Common Difference
- First term (A) = 2005
- Common difference (D) = 2000 - 2005 = -5

### Step 3: Write the General Term Formula
The general term \( T_n \) of an arithmetic progression can be expressed as:
\[ T_n = A + (n - 1)D \]
Substituting the values of A and D:
\[ T_n = 2005 + (n - 1)(-5) \]
This simplifies to:
\[ T_n = 2005 - 5(n - 1) = 2005 - 5n + 5 = 2010 - 5n \]

### Step 4: Set Up the Inequality for Negative Terms
To find the first negative term, we need to determine when \( T_n < 0 \):
\[ 2010 - 5n < 0 \]

### Step 5: Solve the Inequality
Rearranging the inequality:
\[ 2010 < 5n \]
Dividing both sides by 5:
\[ n > \frac{2010}{5} \]
Calculating the right side:
\[ n > 402 \]

### Step 6: Determine the First Integer Value of n
Since n must be a whole number, the smallest integer greater than 402 is 403. Therefore, the first negative term occurs at:
\[ n = 403 \]

### Conclusion
The first negative term of the sequence occurs at the 403rd term.

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