Which term of the A.P.: `33,30,27, ..` is the first negative term?

Answer

To determine which term of the arithmetic progression (A.P.) \(33, 30, 27, \ldots\) is the first negative term, we can follow these steps:

### Step 1: Identify the first term and the common difference
The first term \(a\) of the A.P. is:
\[ a = 33 \]
The common difference \(d\) can be calculated as:
\[ d = \text{second term} - \text{first term} = 30 - 33 = -3 \]

### Step 2: Write the general formula for the n-th term of the A.P.
The n-th term \(a_n\) of an A.P. can be expressed as:
\[ a_n = a + (n - 1) \cdot d \]
Substituting the values of \(a\) and \(d\):
\[ a_n = 33 + (n - 1)(-3) \]

### Step 3: Set up the inequality for the first negative term
To find the first negative term, we need to determine when \(a_n < 0\):
\[ 33 + (n - 1)(-3) < 0 \]
This simplifies to:
\[ 33 - 3(n - 1) < 0 \]
Expanding the equation:
\[ 33 - 3n + 3 < 0 \]
Combining like terms:
\[ 36 - 3n < 0 \]

### Step 4: Solve the inequality
Rearranging the inequality gives:
\[ -3n < -36 \]
Dividing both sides by -3 (remember to reverse the inequality sign):
\[ n > 12 \]

### Step 5: Determine the smallest integer value of \(n\)
Since \(n\) must be greater than 12, the smallest integer value for \(n\) is:
\[ n = 13 \]

### Step 6: Verify that the 13th term is negative
Now, we will check if the 13th term is indeed negative:
\[ a_{13} = 33 + (13 - 1)(-3) = 33 + 12 \cdot (-3) = 33 - 36 = -3 \]
Thus, the 13th term is \(-3\), which is negative.

### Conclusion
The first negative term of the A.P. \(33, 30, 27, \ldots\) is the 13th term.

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