Which term of the A.P. 121, 117, 113,….. Is the first negative terms ?

Answer

To find the first negative term of the arithmetic progression (A.P.) given by 121, 117, 113, ..., we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
- The first term \( a = 121 \).
- The common difference \( d \) can be calculated by subtracting the first term from the second term:
\[ d = 117 - 121 = -4 \]

### Step 2: Write the formula for the nth term of an A.P.
The formula for the nth term \( a_n \) of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]
Substituting the values of \( a \) and \( d \):
\[ a_n = 121 + (n - 1)(-4) \]

### Step 3: Set up the inequality for the first negative term
We want to find the smallest integer \( n \) such that \( a_n < 0 \):
\[ 121 + (n - 1)(-4) < 0 \]

### Step 4: Simplify the inequality
Expanding the inequality:
\[ 121 - 4(n - 1) < 0 \]
This simplifies to:
\[ 121 - 4n + 4 < 0 \]
Combining like terms gives:
\[ 125 - 4n < 0 \]

### Step 5: Solve for n
Rearranging the inequality:
\[ 125 < 4n \]
Dividing both sides by 4:
\[ \frac{125}{4} < n \]
Calculating \( \frac{125}{4} \):
\[ 125 \div 4 = 31.25 \]

### Step 6: Determine the smallest integer n
Since \( n \) must be an integer, we take the next whole number greater than 31.25, which is:
\[ n = 32 \]

### Conclusion
Thus, the first negative term of the A.P. occurs at the 32nd term.