For what value of n , are the `n^(th)` terms of two A.p.s : 63,65,67,`……` and 3,10,17,`……..` equal ?

Answer

To find the value of \( n \) for which the \( n^{th} \) terms of the two arithmetic progressions (APs) are equal, we will follow these steps:

### Step 1: Identify the terms of the two APs
- The first AP is: 63, 65, 67, ...
- The first term \( a_1 \) of the first AP is 63 and the common difference \( d_1 \) is 2.
- The second AP is: 3, 10, 17, ...
- The first term \( a_2 \) of the second AP is 3 and the common difference \( d_2 \) is 7.

### Step 2: Write the formula for the \( n^{th} \) term of an AP
The \( n^{th} \) term of an AP can be calculated using the formula:
\[ T_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 3: Write the \( n^{th} \) terms for both APs
- For the first AP:
\[ T_{n1} = 63 + (n-1) \cdot 2 \]
- For the second AP:
\[ T_{n2} = 3 + (n-1) \cdot 7 \]

### Step 4: Set the \( n^{th} \) terms equal to each other
Set the two expressions for the \( n^{th} \) terms equal:
\[ 63 + (n-1) \cdot 2 = 3 + (n-1) \cdot 7 \]

### Step 5: Expand both sides
Expanding both sides gives:
\[ 63 + 2n - 2 = 3 + 7n - 7 \]
This simplifies to:
\[ 61 + 2n = -4 + 7n \]

### Step 6: Rearrange the equation
Rearranging the equation to isolate \( n \):
\[ 61 + 4 = 7n - 2n \]
\[ 65 = 5n \]

### Step 7: Solve for \( n \)
Now, divide both sides by 5:
\[ n = \frac{65}{5} = 13 \]

### Final Answer
The value of \( n \) for which the \( n^{th} \) terms of the two APs are equal is \( n = 13 \).

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