Question

How many terms of the A.P. 21, 18, 15, ... must be added to get the sum zero?

Hint:

For any AP with a positive first term \( a \) and a negative common difference \( d \), the sum becomes zero when the positive terms are perfectly balanced by the negative terms.
The middle term of such a symmetric sequence must be 0, which occurs here at the 8th term.
Thus, there are 7 positive terms, 1 zero term, and 7 negative terms, giving a total of \( 7 + 1 + 7 = 15 \) terms.

Answer 1

Step 1: Understanding the Question:
The topic of this question is Arithmetic Progressions (AP).
We are given a decreasing arithmetic progression: 21, 18, 15, ...
We need to find the number of terms \( n \) such that their sum \( S_n \) equals zero.

Step 2: Key Formula or Approach:
The sum of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2}[2a + (n - 1)d] \]
We set \( S_n = 0 \) and solve the resulting equation for \( n \). Since \( n \) must be a positive integer, we discard any non-positive or fractional roots.

Step 3: Detailed Explanation:
1. Identify the parameters from the given AP:
- First term, \( a = 21 \)
- Common difference, \( d = 18 - 21 = -3 \)
2. Set the sum formula equal to zero:
\[ S_n = \frac{n}{2}[2a + (n - 1)d] = 0 \]
3. Substitute the values of \( a \) and \( d \):
\[ \frac{n}{2}[2(21) + (n - 1)(-3)] = 0 \]
\[ \frac{n}{2}[42 - 3(n - 1)] = 0 \]
4. Since the number of terms \( n \) must be a positive integer, \( n \neq 0 \). Therefore, we can divide both sides by \( \frac{n}{2} \):
\[ 42 - 3(n - 1) = 0 \]
5. Solve the linear equation for \( n \):
\[ 42 = 3(n - 1) \]
Divide both sides by 3:
\[ 14 = n - 1 \]
Add 1 to both sides:
\[ n = 15 \]
6. Thus, adding the first 15 terms of this AP yields a sum of zero.

Step 4: Final Answer:
Exactly 15 terms of the AP must be added to get a sum of zero.