Find the number of terms of the series 21, 18, 15, 12...which must be taken to give a sum of zero.

Answer

To find the number of terms in the series 21, 18, 15, 12, ... that must be taken to give a sum of zero, we can follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a \) of the series is 21. The common difference \( d \) can be calculated as:
\[ d = \text{second term} - \text{first term} = 18 - 21 = -3 \]

### Step 2: Write the formula for the sum of the first \( n \) terms
The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic series is given by:
\[ S_n = \frac{n}{2} (2a + (n - 1)d) \]
Substituting the values of \( a \) and \( d \):
\[ S_n = \frac{n}{2} (2 \cdot 21 + (n - 1)(-3)) \]
This simplifies to:
\[ S_n = \frac{n}{2} (42 - 3(n - 1)) \]

### Step 3: Simplify the sum expression
Expanding the expression:
\[ S_n = \frac{n}{2} (42 - 3n + 3) = \frac{n}{2} (45 - 3n) \]
Thus:
\[ S_n = \frac{n(45 - 3n)}{2} \]

### Step 4: Set the sum equal to zero
To find the number of terms that gives a sum of zero, we set \( S_n = 0 \):
\[ \frac{n(45 - 3n)}{2} = 0 \]
This implies:
\[ n(45 - 3n) = 0 \]
This gives us two cases:
1. \( n = 0 \) (not valid, as we need a positive number of terms)
2. \( 45 - 3n = 0 \)

### Step 5: Solve for \( n \)
From the second case:
\[ 45 - 3n = 0 \implies 3n = 45 \implies n = \frac{45}{3} = 15 \]

### Conclusion
The number of terms required to give a sum of zero is \( n = 15 \).