Which term of the A.P. 105, 101, 97,………. the first negative term is

Answer

To find which term of the arithmetic progression (A.P.) 105, 101, 97, ... is the first negative term, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the first term (a) and the common difference (d)**:
- The first term \( a = 105 \).
- The common difference \( d \) can be calculated as:
\[ d = \text{second term} - \text{first term} = 101 - 105 = -4 \]

2. **Use the formula for the nth term of an A.P.**:
- The formula for the nth term \( T_n \) of an A.P. is given by:
\[ T_n = a + (n - 1) \cdot d \]
- We want to find the smallest \( n \) for which \( T_n < 0 \).

3. **Set up the equation for the nth term**:
- We set \( T_n = 0 \) to find when the term becomes non-positive:
\[ 0 = 105 + (n - 1)(-4) \]

4. **Solve for n**:
- Rearranging the equation gives:
\[ 0 = 105 - 4(n - 1) \]
\[ 4(n - 1) = 105 \]
\[ n - 1 = \frac{105}{4} = 26.25 \]
\[ n = 26.25 + 1 = 27.25 \]

5. **Determine the first negative term**:
- Since \( n \) must be a whole number, we round up to the next whole number, which is \( n = 28 \).
- Therefore, the first negative term occurs at \( n = 28 \).

### Conclusion:
The first negative term of the A.P. 105, 101, 97, ... is the 28th term.

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