Find the number of terms of the A.P. 98,91,84,…must be taken to give a sum of zero :

Options

A. 14
B. 15
C. 31
D. 29

Answer

To find the number of terms of the arithmetic progression (A.P.) 98, 91, 84, ... 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 A.P. is 98. The common difference \( d \) can be calculated as:
\[ d = 91 - 98 = -7 \]

### Step 2: Write the formula for the sum of the first \( n \) terms of an A.P.
The formula for the sum \( S_n \) of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
Substituting the values of \( a \) and \( d \):
\[ S_n = \frac{n}{2} \times (2 \times 98 + (n - 1)(-7)) \]

### Step 3: Simplify the expression for \( S_n \)
Now, simplify the expression:
\[ S_n = \frac{n}{2} \times (196 - 7(n - 1)) \]
\[ = \frac{n}{2} \times (196 - 7n + 7) \]
\[ = \frac{n}{2} \times (203 - 7n) \]

### Step 4: Set the sum equal to zero
To find the number of terms that give a sum of zero, we set \( S_n = 0 \):
\[ \frac{n}{2} \times (203 - 7n) = 0 \]

### Step 5: Solve the equation
This equation can be satisfied if either \( n = 0 \) or \( 203 - 7n = 0 \). We will solve for \( n \):
1. \( n = 0 \) (not a valid solution in this context)
2. \( 203 - 7n = 0 \)
\[ 7n = 203 \]
\[ n = \frac{203}{7} \approx 29 \]

### Step 6: Conclusion
The number of terms \( n \) that must be taken to give a sum of zero is \( n = 29 \).