Which term of the series 64 + 56 + 48 + ------ is zero

Options

A. (a) 7
B. (b) 9
C. (c) 11
D. (d) 12

Answer

To find which term of the series 64, 56, 48, ... is zero, we first need to determine if the series is an arithmetic progression (AP) and then find the term number when the value becomes zero.

### Step-by-Step Solution:

1. **Identify the first term (a) and common difference (d)**:
- The first term \( a = 64 \).
- To find the common difference \( d \), we subtract the second term from the first term:
\[ d = 56 - 64 = -8 \]

2. **Check if the series is an AP**:
- We need to check if the difference between consecutive terms is constant:
\[ 48 - 56 = -8 \]
- Since both differences are equal, the series is indeed an arithmetic progression.

3. **Use the formula for the nth term of an AP**:
- The nth term of an arithmetic progression can be calculated using the formula:
\[ T_n = a + (n-1) \cdot d \]
- We want to find \( n \) when \( T_n = 0 \):
\[ 0 = 64 + (n-1)(-8) \]

4. **Rearranging the equation**:
- Rearranging gives:
\[ 0 = 64 - 8(n-1) \]
- Simplifying further:
\[ 8(n-1) = 64 \]

5. **Solving for n**:
- Divide both sides by 8:
\[ n-1 = 8 \]
- Adding 1 to both sides:
\[ n = 9 \]

6. **Conclusion**:
- The 9th term of the series is zero.

### Final Answer:
The term of the series that is zero is the **9th term**.

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