The 59th term of an AP is 449 and the 449th term is 59. Which term is equal to 0 (zero)?

Options

A. `501^(st)" term"`
B. `502^(nd)" term"`
C. `508^(th)" term"`
D. `509^(th)" term"`

Explanation

To solve the problem step by step, we will use the properties of an arithmetic progression (AP).

### Step 1: Set up the equations based on the given terms
We know that:
- The 59th term of the AP is given by the formula:
\[ a + (n-1)d \]
For the 59th term (where \( n = 59 \)):
\[ a + 58d = 449 \]
This is our **Equation 1**.

- The 449th term of the AP is given by:
\[ a + (n-1)d \]
For the 449th term (where \( n = 449 \)):
\[ a + 448d = 59 \]
This is our **Equation 2**.

### Step 2: Subtract Equation 2 from Equation 1
Now, we will subtract Equation 2 from Equation 1 to eliminate \( a \):
\[ (a + 58d) - (a + 448d) = 449 - 59 \]
This simplifies to:
\[ 58d - 448d = 390 \]
\[ -390d = 390 \]
Dividing both sides by -390 gives:
\[ d = -1 \]

### Step 3: Substitute \( d \) back to find \( a \)
Now that we have \( d \), we can substitute it back into either Equation 1 or Equation 2 to find \( a \). Let's use Equation 1:
\[ a + 58(-1) = 449 \]
\[ a - 58 = 449 \]
Adding 58 to both sides:
\[ a = 449 + 58 = 507 \]

### Step 4: Find the term that equals zero
We need to find \( n \) such that the \( n \)-th term \( a_n = 0 \):
\[ a_n = a + (n-1)d \]
Setting \( a_n = 0 \):
\[ 0 = 507 + (n-1)(-1) \]
This simplifies to:
\[ 0 = 507 - (n - 1) \]
Rearranging gives:
\[ n - 1 = 507 \]
Thus:
\[ n = 508 \]

### Conclusion
The term that is equal to 0 is the **508th term**.

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