The sum of first four terms of an AP is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms. Also find the sum of all terms of the AP.

Answer

To solve the problem step by step, we will use the properties of an arithmetic progression (AP).

### Step 1: Understand the problem
We are given:
- The first term \( a = 11 \)
- The sum of the first four terms \( S_4 = 56 \)
- The sum of the last four terms \( S_{\text{last 4}} = 112 \)

We need to find the number of terms \( n \) and the total sum of the AP.

### Step 2: Write the formula for the sum of the first four terms
The first four terms of an AP can be expressed as:
- \( a_1 = a \)
- \( a_2 = a + d \)
- \( a_3 = a + 2d \)
- \( a_4 = a + 3d \)

The sum of the first four terms is:
\[ S_4 = a + (a + d) + (a + 2d) + (a + 3d) = 4a + 6d \]
Given \( S_4 = 56 \), we can write:
\[ 4a + 6d = 56 \]

### Step 3: Substitute the value of \( a \)
Substituting \( a = 11 \):
\[ 4(11) + 6d = 56 \]
\[ 44 + 6d = 56 \]
\[ 6d = 56 - 44 \]
\[ 6d = 12 \]
\[ d = 2 \]

### Step 4: Write the formula for the sum of the last four terms
The last four terms can be expressed as:
- \( a_n = a + (n-1)d \)
- \( a_{n-1} = a + (n-2)d \)
- \( a_{n-2} = a + (n-3)d \)
- \( a_{n-3} = a + (n-4)d \)

The sum of the last four terms is:
\[ S_{\text{last 4}} = a_n + a_{n-1} + a_{n-2} + a_{n-3} = 4a + (n-10)d \]
Given \( S_{\text{last 4}} = 112 \), we can write:
\[ 4a + (n-10)d = 112 \]

### Step 5: Substitute the values of \( a \) and \( d \)
Substituting \( a = 11 \) and \( d = 2 \):
\[ 4(11) + (n-10)(2) = 112 \]
\[ 44 + 2(n-10) = 112 \]
\[ 44 + 2n - 20 = 112 \]
\[ 2n + 24 = 112 \]
\[ 2n = 112 - 24 \]
\[ 2n = 88 \]
\[ n = 44 \]

### Step 6: Calculate the sum of all terms in the AP
The sum \( S_n \) of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Substituting \( n = 44 \), \( a = 11 \), and \( d = 2 \):
\[ S_{44} = \frac{44}{2} \times (2(11) + (44-1)(2)) \]
\[ = 22 \times (22 + 86) \]
\[ = 22 \times 108 \]
\[ = 2376 \]

### Final Answers
- The number of terms \( n = 44 \)
- The sum of all terms \( S_n = 2376 \)