Find the middle term of an A.P. `-6, -2, 2,`……….58.

Answer

To find the middle term of the arithmetic progression (A.P.) given by the series `-6, -2, 2, ..., 58`, we will follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a \) of the A.P. is:
\[ a = -6 \]

To find the common difference \( d \), we subtract the first term from the second term:
\[ d = -2 - (-6) = -2 + 6 = 4 \]

### Step 2: Use the formula for the nth term of an A.P.
The formula for the nth term \( a_n \) of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]
We know that the last term \( a_n \) is 58. Therefore, we can set up the equation:
\[ 58 = -6 + (n - 1) \cdot 4 \]

### Step 3: Solve for \( n \)
First, we simplify the equation:
\[ 58 + 6 = (n - 1) \cdot 4 \]
\[ 64 = (n - 1) \cdot 4 \]

Now, divide both sides by 4:
\[ n - 1 = \frac{64}{4} \]
\[ n - 1 = 16 \]

Adding 1 to both sides gives:
\[ n = 17 \]

### Step 4: Find the middle term
Since there are 17 terms in total, the middle term is the \( \frac{n + 1}{2} \)th term:
\[ \text{Middle term position} = \frac{17 + 1}{2} = 9 \]

### Step 5: Find the 9th term
Now we need to find the 9th term \( a_9 \):
Using the formula for the nth term:
\[ a_9 = a + (9 - 1) \cdot d \]
Substituting the values we have:
\[ a_9 = -6 + (8) \cdot 4 \]
\[ a_9 = -6 + 32 \]
\[ a_9 = 26 \]

### Conclusion
The middle term of the given A.P. is:
\[ \boxed{26} \]