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 step-by-step, we will use the properties of Arithmetic Progressions (AP).

### Step 1: Define the terms of the two APs
Let the first AP be represented by its first term \( A_1 \) and the second AP by its first term \( A_2 \). Both APs have the same common difference \( D \).

### Step 2: Write the formula for the \( n^{th} \) term of an AP
The \( n^{th} \) term of an AP can be expressed as:
\[ A_n = A + (n-1)D \]
Thus, the \( 50^{th} \) term of the first AP (\( A_1 \)) is:
\[ A_{1,50} = A_1 + (50-1)D = A_1 + 49D \]
And the \( 50^{th} \) term of the second AP (\( A_2 \)) is:
\[ A_{2,50} = A_2 + (50-1)D = A_2 + 49D \]

### Step 3: Set up the equation for the difference between the \( 50^{th} \) terms
According to the problem, the difference between the \( 50^{th} \) terms of the two APs is 100:
\[ (A_1 + 49D) - (A_2 + 49D) = 100 \]
This simplifies to:
\[ A_1 - A_2 = 100 \]

### Step 4: Write the formula for the \( 100^{th} \) term of both APs
Now, we will find the \( 100^{th} \) term of the first AP:
\[ A_{1,100} = A_1 + (100-1)D = A_1 + 99D \]
And for the second AP:
\[ A_{2,100} = A_2 + (100-1)D = A_2 + 99D \]

### Step 5: Set up the equation for the difference between the \( 100^{th} \) terms
Now, we calculate the difference between the \( 100^{th} \) terms:
\[ (A_1 + 99D) - (A_2 + 99D) = A_1 - A_2 \]
Since we already found that \( A_1 - A_2 = 100 \), we can substitute this value:
\[ A_{1,100} - A_{2,100} = 100 \]

### Conclusion
Thus, the difference between the \( 100^{th} \) terms of the two APs is also \( 100 \).

### Final Answer
The difference between their \( 100^{th} \) terms is **100**.

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