For what value of n, the nth terms of the arithmetic progressions 63, 65, 67, … and 3, 10, 17,… are equal?

Explanation

To find the value of \( n \) for which the \( n \)th terms of the arithmetic progressions (APs) 63, 65, 67, ... and 3, 10, 17, ... are equal, we can follow these steps:

### Step 1: Identify the first term and common difference of both APs.

For the first AP (63, 65, 67,...):
- First term \( a_1 = 63 \)
- Common difference \( d_1 = 65 - 63 = 2 \)

For the second AP (3, 10, 17,...):
- First term \( a_2 = 3 \)
- Common difference \( d_2 = 10 - 3 = 7 \)

### Step 2: Write the formula for the \( n \)th term of each AP.

The \( n \)th term of an AP is given by the formula:
\[ T_n = a + (n - 1) \cdot d \]

For the first AP:
\[ T_n^{(1)} = 63 + (n - 1) \cdot 2 \]

For the second AP:
\[ T_n^{(2)} = 3 + (n - 1) \cdot 7 \]

### Step 3: Set the two \( n \)th terms equal to each other.

We need to find \( n \) such that:
\[ 63 + (n - 1) \cdot 2 = 3 + (n - 1) \cdot 7 \]

### Step 4: Simplify the equation.

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

### Step 5: Rearrange the equation to isolate \( n \).

Rearranging gives:
\[ 61 + 4 = 7n - 2n \]
\[ 65 = 5n \]

### 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 the two arithmetic progressions are equal is \( n = 13 \).

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