For what value of n, the nth term of A.P. 63, 65, 67,……... and nth term of A.P. 3. 10. 17. ...... are equal to each other?

Answer

To find the value of \( n \) for which the \( n \)th term of the first A.P. (63, 65, 67, ...) is equal to the \( n \)th term of the second A.P. (3, 10, 17, ...), we will use the formula for the \( n \)th term of an arithmetic progression (A.P.), which is given by:

\[ T_n = a + (n - 1) \cdot d \]

where \( a \) is the first term and \( d \) is the common difference.

### Step 1: Identify the first A.P.
- First A.P.: 63, 65, 67, ...
- First term \( a_1 = 63 \)
- Common difference \( d_1 = 65 - 63 = 2 \)

The \( n \)th term of the first A.P. can be expressed as:
\[ T_n^{(1)} = 63 + (n - 1) \cdot 2 \]

### Step 2: Identify the second A.P.
- Second A.P.: 3, 10, 17, ...
- First term \( a_2 = 3 \)
- Common difference \( d_2 = 10 - 3 = 7 \)

The \( n \)th term of the second A.P. can be expressed as:
\[ T_n^{(2)} = 3 + (n - 1) \cdot 7 \]

### Step 3: Set the two \( n \)th terms equal to each other
We need to find \( n \) such that:
\[ T_n^{(1)} = T_n^{(2)} \]
This gives us the equation:
\[ 63 + (n - 1) \cdot 2 = 3 + (n - 1) \cdot 7 \]

### Step 4: Simplify the equation
Expanding both sides:
\[ 63 + 2n - 2 = 3 + 7n - 7 \]
This simplifies to:
\[ 61 + 2n = -4 + 7n \]

### Step 5: Rearranging the equation
Now, we will rearrange the equation to isolate \( n \):
\[ 61 + 2n + 4 = 7n \]
\[ 65 = 7n - 2n \]
\[ 65 = 5n \]

### Step 6: Solve for \( n \)
Dividing both sides by 5:
\[ n = \frac{65}{5} = 13 \]

### Conclusion
The value of \( n \) for which the \( n \)th terms of both A.P.s are equal is:
\[ \boxed{13} \]

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