Which term in the A.P. 5,2,-1,… is -22 ?

Answer

To find which term in the arithmetic progression (A.P.) 5, 2, -1, ... is equal to -22, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the first term (a) and the common difference (d)**:
- The first term \( a = 5 \).
- To find the common difference \( d \), we subtract the first term from the second term:
\[ d = 2 - 5 = -3 \]
- We can verify this by checking the difference between the second and third terms:
\[ d = -1 - 2 = -3 \]
- Thus, the common difference \( d = -3 \).

2. **Write the formula for the nth term of an A.P.**:
- The nth term \( T_n \) of an A.P. can be expressed as:
\[ T_n = a + (n - 1) \cdot d \]

3. **Set up the equation for the nth term equal to -22**:
- We need to find \( n \) such that:
\[ T_n = -22 \]
- Substituting the values of \( a \) and \( d \) into the formula:
\[ -22 = 5 + (n - 1)(-3) \]

4. **Simplify the equation**:
- Rearranging the equation gives:
\[ -22 = 5 - 3(n - 1) \]
- This simplifies to:
\[ -22 = 5 - 3n + 3 \]
- Combining like terms:
\[ -22 = 8 - 3n \]

5. **Isolate the term involving n**:
- Move 8 to the left side:
\[ -22 - 8 = -3n \]
\[ -30 = -3n \]

6. **Solve for n**:
- Dividing both sides by -3:
\[ n = \frac{-30}{-3} = 10 \]

7. **Conclusion**:
- Therefore, the term in the A.P. that is equal to -22 is the **10th term**.