If the sum of the first 23 terms of an arithmetic progression equals that of the first 37 terms , then the sum of the first 60 terms equals .

Answer

To solve the problem step by step, we will use the formula for the sum of the first n terms of an arithmetic progression (AP). The formula is given by:

\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]

where:
- \( S_n \) is the sum of the first n terms,
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the number of terms.

### Step 1: Set up the equations for the sums

We are given that the sum of the first 23 terms equals the sum of the first 37 terms. Therefore, we can write:

\[ S_{23} = S_{37} \]

Using the formula for the sum of an AP, we have:

\[ S_{23} = \frac{23}{2} \times (2a + (23-1)d) = \frac{23}{2} \times (2a + 22d) \]

\[ S_{37} = \frac{37}{2} \times (2a + (37-1)d) = \frac{37}{2} \times (2a + 36d) \]

### Step 2: Equate the two sums

Setting \( S_{23} \) equal to \( S_{37} \):

\[ \frac{23}{2} \times (2a + 22d) = \frac{37}{2} \times (2a + 36d) \]

### Step 3: Simplify the equation

We can eliminate the \(\frac{1}{2}\) from both sides:

\[ 23(2a + 22d) = 37(2a + 36d) \]

### Step 4: Expand both sides

Expanding both sides gives:

\[ 46a + 506d = 74a + 1332d \]

### Step 5: Rearrange the equation

Rearranging the equation to isolate terms involving \( a \) and \( d \):

\[ 46a - 74a + 506d - 1332d = 0 \]

This simplifies to:

\[ -28a - 826d = 0 \]

### Step 6: Solve for \( a \) in terms of \( d \)

Rearranging gives:

\[ 28a + 826d = 0 \]

Dividing through by 14:

\[ 2a + 59d = 0 \]

### Step 7: Find the sum of the first 60 terms

Now we need to find the sum of the first 60 terms \( S_{60} \):

\[ S_{60} = \frac{60}{2} \times (2a + (60-1)d) = 30 \times (2a + 59d) \]

### Step 8: Substitute the value of \( 2a + 59d \)

From our previous equation, we know that:

\[ 2a + 59d = 0 \]

Thus, substituting this into the equation for \( S_{60} \):

\[ S_{60} = 30 \times 0 = 0 \]

### Final Answer

The sum of the first 60 terms of the arithmetic progression is:

\[ \boxed{0} \]