A geometrical progression of positive terms and an arithmetical progression have the same first term. The sum of their first terms is 1 , the sum of their second terms is `(1)/(2)` and the sum of their third terms is 2. Calculate the sum of their fourth terms.

Answer

To solve the problem step by step, we will define the terms of the arithmetic progression (AP) and geometric progression (GP) and then use the given conditions to find the required sum of their fourth terms.

### Step 1: Define the terms of AP and GP
Let the first term of both the AP and GP be \( a \).
- The terms of the AP are: \( a, a + d, a + 2d, a + 3d, \ldots \)
- The terms of the GP are: \( a, ar, ar^2, ar^3, \ldots \)

### Step 2: Set up the equations based on the given conditions
1. The sum of their first terms:
\[ a + a = 1 \quad \Rightarrow \quad 2a = 1 \quad \Rightarrow \quad a = \frac{1}{2} \]

2. The sum of their second terms:
\[ (a + d) + ar = \frac{1}{2} \]
Substituting \( a = \frac{1}{2} \):
\[ \left(\frac{1}{2} + d\right) + \frac{1}{2}r = \frac{1}{2} \]
Simplifying this:
\[ d + \frac{1}{2}r = 0 \quad \Rightarrow \quad d = -\frac{1}{2}r \]

3. The sum of their third terms:
\[ (a + 2d) + ar^2 = 2 \]
Substituting \( a = \frac{1}{2} \) and \( d = -\frac{1}{2}r \):
\[ \left(\frac{1}{2} + 2\left(-\frac{1}{2}r\right)\right) + \frac{1}{2}r^2 = 2 \]
Simplifying this:
\[ \frac{1}{2} - r + \frac{1}{2}r^2 = 2 \]
Multiplying through by 2 to eliminate the fraction:
\[ 1 - 2r + r^2 = 4 \quad \Rightarrow \quad r^2 - 2r - 3 = 0 \]

### Step 3: Solve the quadratic equation
The quadratic equation \( r^2 - 2r - 3 = 0 \) can be factored:
\[ (r - 3)(r + 1) = 0 \]
Thus, the solutions for \( r \) are:
\[ r = 3 \quad \text{(positive term)} \quad \text{or} \quad r = -1 \quad \text{(neglected)} \]

### Step 4: Find \( d \)
Substituting \( r = 3 \) back into the equation for \( d \):
\[ d = -\frac{1}{2}r = -\frac{1}{2}(3) = -\frac{3}{2} \]

### Step 5: Calculate the fourth terms
The fourth term of the AP is:
\[ a + 3d = \frac{1}{2} + 3\left(-\frac{3}{2}\right) = \frac{1}{2} - \frac{9}{2} = -\frac{8}{2} = -4 \]

The fourth term of the GP is:
\[ ar^3 = \frac{1}{2}(3^3) = \frac{1}{2}(27) = \frac{27}{2} \]

### Step 6: Calculate the sum of the fourth terms
Now, we sum the fourth terms:
\[ \text{Sum of fourth terms} = -4 + \frac{27}{2} \]
Converting \(-4\) to a fraction:
\[ -4 = -\frac{8}{2} \]
Thus,
\[ \text{Sum} = -\frac{8}{2} + \frac{27}{2} = \frac{19}{2} \]

### Final Answer
The sum of their fourth terms is:
\[ \frac{19}{2} \]