If the 8th term of an A.P. is 37 and the 15th term is 15 more than the 12th find the A.P. Also find the sum of first 20 terms of this A.P.

Answer

To solve the problem step by step, we will follow the information given about the arithmetic progression (A.P.) and use the formulas related to it.

### Step 1: Understand the terms of A.P.
The nth term of an A.P. can be expressed as:
\[ T_n = A + (n-1)D \]
where:
- \( T_n \) is the nth term,
- \( A \) is the first term,
- \( D \) is the common difference.

### Step 2: Write the equations based on the problem statement.
Given:
- The 8th term \( T_8 = 37 \)
- The 15th term \( T_{15} \) is 15 more than the 12th term \( T_{12} \).

From the first condition:
\[ T_8 = A + 7D = 37 \quad \text{(1)} \]

From the second condition:
\[ T_{15} = A + 14D \]
\[ T_{12} = A + 11D \]
Thus, we can write:
\[ A + 14D = A + 11D + 15 \]
This simplifies to:
\[ 14D - 11D = 15 \]
\[ 3D = 15 \]
So, we find:
\[ D = 5 \quad \text{(2)} \]

### Step 3: Substitute \( D \) back into equation (1).
Now substitute \( D = 5 \) into equation (1):
\[ A + 7(5) = 37 \]
\[ A + 35 = 37 \]
Subtracting 35 from both sides gives:
\[ A = 2 \quad \text{(3)} \]

### Step 4: Write down the A.P.
Now that we have \( A \) and \( D \):
- First term \( A = 2 \)
- Common difference \( D = 5 \)

The A.P. can be written as:
1st term: \( 2 \)
2nd term: \( 2 + 5 = 7 \)
3rd term: \( 2 + 2(5) = 12 \)
4th term: \( 2 + 3(5) = 17 \)
5th term: \( 2 + 4(5) = 22 \)
6th term: \( 2 + 5(5) = 27 \)
7th term: \( 2 + 6(5) = 32 \)
8th term: \( 2 + 7(5) = 37 \)
9th term: \( 2 + 8(5) = 42 \)
10th term: \( 2 + 9(5) = 47 \)
11th term: \( 2 + 10(5) = 52 \)
12th term: \( 2 + 11(5) = 57 \)
13th term: \( 2 + 12(5) = 62 \)
14th term: \( 2 + 13(5) = 67 \)
15th term: \( 2 + 14(5) = 72 \)
16th term: \( 2 + 15(5) = 77 \)
17th term: \( 2 + 16(5) = 82 \)
18th term: \( 2 + 17(5) = 87 \)
19th term: \( 2 + 18(5) = 92 \)
20th term: \( 2 + 19(5) = 97 \)

So, the A.P. is:
\[ 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97 \]

### Step 5: Find the sum of the first 20 terms of the A.P.
The sum of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2A + (n-1)D) \]

For \( n = 20 \):
\[ S_{20} = \frac{20}{2} \times (2 \times 2 + (20 - 1) \times 5) \]
\[ = 10 \times (4 + 19 \times 5) \]
\[ = 10 \times (4 + 95) \]
\[ = 10 \times 99 \]
\[ = 990 \]

### Final Answer:
The A.P. is \( 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97 \) and the sum of the first 20 terms is \( 990 \).