In an A.P., If a=21 , d=-3 and `a_(n)=0` , then find the value of 'n' .

Answer

To solve the problem step by step, we will use the formula for the nth term of an arithmetic progression (A.P.).

### Step 1: Write the formula for the nth term of an A.P.
The nth term \( a_n \) of an A.P. can be expressed as:
\[ a_n = a + (n - 1) \cdot d \]
where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

### Step 2: Substitute the given values into the formula
We are given:
- \( a = 21 \)
- \( d = -3 \)
- \( a_n = 0 \)

Substituting these values into the formula gives us:
\[ 0 = 21 + (n - 1)(-3) \]

### Step 3: Simplify the equation
Now, we will simplify the equation:
\[ 0 = 21 - 3(n - 1) \]
This can be rearranged to:
\[ 0 = 21 - 3n + 3 \]
Combining like terms results in:
\[ 0 = 24 - 3n \]

### Step 4: Isolate the term involving \( n \)
Next, we will isolate the term involving \( n \) by moving 24 to the other side:
\[ 3n = 24 \]

### Step 5: Solve for \( n \)
Now, divide both sides by 3 to find \( n \):
\[ n = \frac{24}{3} = 8 \]

### Conclusion
Thus, the value of \( n \) is:
\[ \boxed{8} \]

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