The common difference between the terms of two AP's is same. If the difference between their `50^(th)` terms is 100, what is the difference between their `100^(th)` terms?

Answer

To solve the problem, we will use the properties of Arithmetic Progressions (AP). Let's denote the first AP as \( A_n \) and the second AP as \( T_n \).

### Step-by-Step Solution:

1. **Understand the nth Term Formula**:
The nth term of an AP can be expressed as:
\[ A_n = A_1 + (n-1)D \]
where \( A_1 \) is the first term and \( D \) is the common difference.

2. **Write the 50th Terms**:
For the first AP:
\[ A_{50} = A_1 + (50-1)D = A_1 + 49D \]
For the second AP:
\[ T_{50} = T_1 + (50-1)D = T_1 + 49D \]

3. **Set Up the Difference Between the 50th Terms**:
According to the problem, the difference between the 50th terms of the two APs is given as:
\[ A_{50} - T_{50} = 100 \]
Substituting the expressions for \( A_{50} \) and \( T_{50} \):
\[ (A_1 + 49D) - (T_1 + 49D) = 100 \]
Simplifying this gives:
\[ A_1 - T_1 = 100 \]

4. **Write the 100th Terms**:
Now, we will write the 100th terms for both APs.
For the first AP:
\[ A_{100} = A_1 + (100-1)D = A_1 + 99D \]
For the second AP:
\[ T_{100} = T_1 + (100-1)D = T_1 + 99D \]

5. **Set Up the Difference Between the 100th Terms**:
We need to find the difference between the 100th terms:
\[ A_{100} - T_{100} = (A_1 + 99D) - (T_1 + 99D) \]
Simplifying this gives:
\[ A_{100} - T_{100} = A_1 - T_1 \]

6. **Substitute the Value of \( A_1 - T_1 \)**:
From step 3, we found that \( A_1 - T_1 = 100 \). Thus:
\[ A_{100} - T_{100} = 100 \]

### Final Answer:
The difference between the 100th terms of the two APs is **100**.