The sum of the first fifteen terms of an arithmetical progression is 105 and the sum of the next fifteen terms is 780. Find the first three terms of the arithmetical progression,.

Answer

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

### Step 1: Use the formula for the sum of the first n terms of an AP.
The formula for the sum of the first n terms \( S_n \) of an arithmetic progression is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the number of terms.

### Step 2: Set up the equation for the first 15 terms.
Given that the sum of the first 15 terms is 105, we can write:
\[ S_{15} = \frac{15}{2} \times (2a + (15-1)d) = 105 \]
This simplifies to:
\[ \frac{15}{2} \times (2a + 14d) = 105 \]
Multiplying both sides by 2 to eliminate the fraction:
\[ 15 \times (2a + 14d) = 210 \]
Dividing both sides by 15:
\[ 2a + 14d = 14 \quad \text{(Equation 1)} \]

### Step 3: Set up the equation for the next 15 terms.
The sum of the next 15 terms is given as 780. The sum of the first 30 terms can be expressed as:
\[ S_{30} = S_{15} + S_{next\ 15} \]
Thus:
\[ S_{30} - S_{15} = 780 \]
Substituting \( S_{15} = 105 \):
\[ S_{30} - 105 = 780 \]
This gives:
\[ S_{30} = 885 \]
Now, using the formula for \( S_{30} \):
\[ S_{30} = \frac{30}{2} \times (2a + (30-1)d) = 885 \]
This simplifies to:
\[ 15 \times (2a + 29d) = 885 \]
Dividing both sides by 15:
\[ 2a + 29d = 59 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations.
Now we have two equations:
1. \( 2a + 14d = 14 \) (Equation 1)
2. \( 2a + 29d = 59 \) (Equation 2)

To eliminate \( 2a \), we can subtract Equation 1 from Equation 2:
\[ (2a + 29d) - (2a + 14d) = 59 - 14 \]
This simplifies to:
\[ 15d = 45 \]
Dividing both sides by 15:
\[ d = 3 \]

### Step 5: Substitute \( d \) back to find \( a \).
Now substitute \( d = 3 \) back into Equation 1:
\[ 2a + 14(3) = 14 \]
This simplifies to:
\[ 2a + 42 = 14 \]
Subtracting 42 from both sides:
\[ 2a = 14 - 42 \]
\[ 2a = -28 \]
Dividing by 2:
\[ a = -14 \]

### Step 6: Find the first three terms of the AP.
Now that we have \( a \) and \( d \), we can find the first three terms:
- First term \( a = -14 \)
- Second term \( a + d = -14 + 3 = -11 \)
- Third term \( a + 2d = -14 + 2(3) = -14 + 6 = -8 \)

### Final Answer:
The first three terms of the arithmetic progression are:
\[ -14, -11, -8 \]