Which term of the AP: 115, 110, 105,.............is its first negative term?

Answer

To find the first negative term of the arithmetic progression (AP) given by the sequence 115, 110, 105, ..., we will follow these steps:

### Step 1: Identify the first term and common difference
- The first term \( a \) of the AP is 115.
- The common difference \( d \) can be calculated as:
\[ d = 110 - 115 = -5 \]

### Step 2: Use the formula for the nth term of an AP
The formula for the nth term \( a_n \) of an arithmetic progression is given by:
\[ a_n = a + (n - 1)d \]
where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

### Step 3: Set up the equation for the first negative term
We want to find the first term that is negative. Hence, we set \( a_n < 0 \). We can find the term where \( a_n = 0 \) first, and then the next term will be the first negative term.

Setting \( a_n = 0 \):
\[ 0 = 115 + (n - 1)(-5) \]

### Step 4: Solve for \( n \)
Rearranging the equation:
\[ 0 = 115 - 5(n - 1) \]
\[ 5(n - 1) = 115 \]
\[ n - 1 = \frac{115}{5} \]
\[ n - 1 = 23 \]
\[ n = 24 \]

### Step 5: Determine the first negative term
Since the 24th term is 0, the next term (25th term) will be:
\[ a_{25} = 115 + (25 - 1)(-5) = 115 - 120 = -5 \]

### Conclusion
Thus, the first negative term of the AP occurs at the 25th position.

**Final Answer: The first negative term is the 25th term.**

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