Find the 25th term of the A.P. 10,6,2,-2,-6 ,-10 ,… :

Options

A. `-86`
B. 106
C. 96
D. none of these

Answer

To find the 25th term of the arithmetic progression (A.P.) given by the sequence 10, 6, 2, -2, -6, -10, ..., we will follow these steps:

### Step-by-Step Solution:

1. **Identify the First Term (a)**:
The first term of the A.P. is given as:
\[ a = 10 \]

2. **Calculate the Common Difference (d)**:
The common difference \(d\) can be calculated by subtracting the first term from the second term:
\[ d = T_2 - T_1 = 6 - 10 = -4 \]

3. **Use the Formula for the nth Term of an A.P.**:
The formula for the nth term of an A.P. is given by:
\[ T_n = a + (n - 1) \cdot d \]
Here, we need to find the 25th term, so \(n = 25\).

4. **Substitute the Values into the Formula**:
Substitute \(a = 10\), \(d = -4\), and \(n = 25\) into the formula:
\[ T_{25} = 10 + (25 - 1) \cdot (-4) \]
Simplifying this:
\[ T_{25} = 10 + 24 \cdot (-4) \]

5. **Calculate the Value**:
Now calculate \(24 \cdot (-4)\):
\[ 24 \cdot (-4) = -96 \]
Therefore:
\[ T_{25} = 10 - 96 = -86 \]

### Final Answer:
The 25th term of the A.P. is:
\[ \boxed{-86} \]