The `6^(th)` term from the end of the A.P. 5, 2, -1, -4, …., -31, is

Options

A. `-25`
B. `-22`
C. `-19`
D. `-16`

Answer

To find the 6th term from the end of the arithmetic progression (A.P.) given by the sequence 5, 2, -1, -4, ..., -31, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the first term (a) and the common difference (d)**:
- The first term \( a = 5 \)
- The common difference \( d = 2 - 5 = -3 \)

2. **Determine the last term (Tn)**:
- The last term given in the sequence is \( T_n = -31 \).

3. **Find the number of terms (n) in the A.P.**:
- The formula for the nth term of an A.P. is:
\[ T_n = a + (n-1) \cdot d \]
- Substituting the known values:
\[ -31 = 5 + (n-1)(-3) \]
- Rearranging the equation:
\[ -31 - 5 = (n-1)(-3) \]
\[ -36 = (n-1)(-3) \]
- Dividing both sides by -3:
\[ n - 1 = 12 \]
- Therefore, \( n = 13 \).

4. **Calculate the 6th term from the end**:
- The 6th term from the end can be found using the formula:
\[ T_{n-5} = a + (n-5-1) \cdot d \]
- Substituting \( n = 13 \):
\[ T_{13-5} = a + (13-5-1) \cdot d \]
\[ T_{8} = 5 + (7)(-3) \]
\[ T_{8} = 5 - 21 \]
\[ T_{8} = -16 \]

5. **Conclusion**:
- The 6th term from the end of the A.P. is \( -16 \).

### Final Answer:
The 6th term from the end of the A.P. is \( -16 \).

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