Which term of the series : 21, 18, 15, is -81 ?
Can any term this series be zero ? If yes, find the number of terms.

Answer

To solve the problem step by step, we will first determine which term of the series 21, 18, 15 is -81, and then we will check if any term in this series can be zero.

### Step 1: Identify the first term (a) and common difference (d)
The series is an arithmetic progression (AP) where:
- The first term \( a = 21 \)
- The second term \( T_2 = 18 \)

To find the common difference \( d \):
\[ d = T_2 - T_1 = 18 - 21 = -3 \]

**Hint:** The common difference is found by subtracting the first term from the second term.

### Step 2: Use the formula for the nth term of an AP
The formula for the nth term \( T_n \) of an arithmetic progression is given by:
\[ T_n = a + (n-1)d \]
We need to find \( n \) such that \( T_n = -81 \).

**Hint:** The nth term formula helps us express any term in the series based on its position.

### Step 3: Set up the equation
Substituting the known values into the formula:
\[ -81 = 21 + (n-1)(-3) \]

**Hint:** Substitute the values of \( a \) and \( d \) into the nth term formula.

### Step 4: Simplify the equation
Rearranging the equation gives:
\[ -81 = 21 - 3(n - 1) \]
\[ -81 - 21 = -3(n - 1) \]
\[ -102 = -3(n - 1) \]

**Hint:** Move all terms involving \( n \) to one side and constants to the other side.

### Step 5: Solve for \( n \)
Dividing both sides by -3:
\[ n - 1 = \frac{102}{3} \]
\[ n - 1 = 34 \]
\[ n = 35 \]

**Hint:** When dividing by a negative number, remember to keep the direction of the inequality in mind.

### Step 6: Conclusion for the first part
Thus, the term -81 is the 35th term of the series.

### Step 7: Check if any term can be zero
Now, we need to check if any term in this series can be zero. We set \( T_m = 0 \):
\[ 0 = 21 + (m-1)(-3) \]

**Hint:** Use the same nth term formula to find when a term equals zero.

### Step 8: Set up the equation for zero
Rearranging gives:
\[ 0 = 21 - 3(m - 1) \]
\[ 3(m - 1) = 21 \]
\[ m - 1 = 7 \]
\[ m = 8 \]

**Hint:** Isolate \( m \) to find the position of the term that equals zero.

### Step 9: Conclusion for the second part
Thus, the 8th term of the series is 0.

### Final Summary
1. The term -81 is the 35th term of the series.
2. Yes, there is a term that is 0, and it is the 8th term.