Is -150 a term of 11, 8, 5, 2…………

Answer

To determine whether -150 is a term of the arithmetic progression (AP) given by the sequence 11, 8, 5, 2, ..., we can follow these steps:

### Step-by-Step Solution:

1. **Identify the first term (a) and the common difference (d)**:
- The first term \( a = 11 \).
- The common difference \( d \) can be calculated as:
\[ d = 8 - 11 = -3 \]

2. **Use the formula for the n-th term of an AP**:
- The n-th term \( T_n \) of an AP can be expressed as:
\[ T_n = a + (n-1)d \]
- We want to find out if -150 can be expressed as \( T_n \):
\[ -150 = 11 + (n-1)(-3) \]

3. **Rearranging the equation**:
- Rearranging the equation gives:
\[ -150 - 11 = (n-1)(-3) \]
\[ -161 = (n-1)(-3) \]

4. **Solving for n**:
- Dividing both sides by -3:
\[ n-1 = \frac{-161}{-3} = \frac{161}{3} \]
- Therefore:
\[ n = \frac{161}{3} + 1 = \frac{161}{3} + \frac{3}{3} = \frac{164}{3} \]

5. **Check if n is a whole number**:
- The value \( \frac{164}{3} \) is not a whole number (it equals approximately 54.67). Since n must be a positive integer, -150 cannot be a term of the AP.

6. **Conclusion**:
- Since n is not an integer, we conclude that -150 is not a term of the arithmetic progression.

### Final Answer:
- **-150 is not a term of the arithmetic progression 11, 8, 5, 2, ...**.

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