If the nth term of the A.P. 58, 60, 62, is equal to the nth term of the A.P. -2, 5, 12………….. find the value of n.

Answer

To solve the problem, we need to find the value of \( n \) such that the \( n \)th term of the first arithmetic progression (A.P.) is equal to the \( n \)th term of the second A.P.

### Step 1: Identify the first A.P.
The first A.P. is given as:
58, 60, 62, ...

- The first term \( a_1 = 58 \)
- The common difference \( d_1 = 60 - 58 = 2 \)

### Step 2: Write the formula for the \( n \)th term of the first A.P.
The \( n \)th term \( T_n^{(1)} \) of the first A.P. can be calculated using the formula:
\[ T_n^{(1)} = a_1 + (n - 1) d_1 \]
Substituting the values:
\[ T_n^{(1)} = 58 + (n - 1) \cdot 2 \]
Simplifying this:
\[ T_n^{(1)} = 58 + 2n - 2 = 56 + 2n \]

### Step 3: Identify the second A.P.
The second A.P. is given as:
-2, 5, 12, ...

- The first term \( a_2 = -2 \)
- The common difference \( d_2 = 5 - (-2) = 7 \)

### Step 4: Write the formula for the \( n \)th term of the second A.P.
The \( n \)th term \( T_n^{(2)} \) of the second A.P. can be calculated using the formula:
\[ T_n^{(2)} = a_2 + (n - 1) d_2 \]
Substituting the values:
\[ T_n^{(2)} = -2 + (n - 1) \cdot 7 \]
Simplifying this:
\[ T_n^{(2)} = -2 + 7n - 7 = 7n - 9 \]

### Step 5: Set the two \( n \)th terms equal to each other.
Since the \( n \)th terms of both A.P.s are equal, we have:
\[ T_n^{(1)} = T_n^{(2)} \]
This gives us the equation:
\[ 56 + 2n = 7n - 9 \]

### Step 6: Solve for \( n \).
Rearranging the equation:
\[ 56 + 9 = 7n - 2n \]
\[ 65 = 5n \]
Dividing both sides by 5:
\[ n = \frac{65}{5} = 13 \]

### Conclusion
The value of \( n \) is \( 13 \).

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