Check whether `-150` is a term of the A.P: 11,8,5,2`…..`

Answer

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

### Step-by-Step Solution:

1. **Identify the first term (a)**:
The first term of the A.P. is given as:
\[ a = 11 \]

2. **Calculate the common difference (d)**:
The common difference can be calculated by subtracting the first term from the second term:
\[ d = 8 - 11 = -3 \]

3. **Use the formula for the n-th term of an A.P.**:
The formula for the n-th term of an A.P. is given by:
\[ a_n = a + (n - 1)d \]
We need to check if `-150` can be expressed as \( a_n \).

4. **Set up the equation**:
We will set \( a_n \) equal to `-150`:
\[ -150 = 11 + (n - 1)(-3) \]

5. **Simplify the equation**:
Rearranging the equation gives:
\[ -150 = 11 - 3(n - 1) \]
\[ -150 = 11 - 3n + 3 \]
\[ -150 = 14 - 3n \]

6. **Isolate n**:
Now, we will isolate \( n \):
\[ -150 - 14 = -3n \]
\[ -164 = -3n \]
Dividing both sides by -3 gives:
\[ n = \frac{164}{3} \]

7. **Check if n is a whole number**:
Calculating \( \frac{164}{3} \):
\[ n = 54.67 \]
Since \( n \) is not a whole number, it indicates that `-150` is not a term of the A.P.

### Conclusion:
Since \( n \) is not a whole number, we conclude that `-150` is **not** a term of the arithmetic progression `11, 8, 5, 2, ...`.

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