There are four numbers such that the first three of them form an Arithmetic Progression and the last three form a Geometric Progression. The sum of the first and the third is 2. and that of second and fourth is 26. What is the sum of first and fourth number ?

Options

A. a)`85(13)/(25)`
B. b)`25(73)/(85)`
C. c)83
D. d)none of these

Answer

To solve the problem, we need to find the sum of the first and fourth numbers given certain conditions about the sequences they form. Let's denote the four numbers as \( A, B, C, D \).

### Step 1: Set up the equations based on the problem statement
1. The first three numbers \( A, B, C \) form an Arithmetic Progression (AP).
2. The last three numbers \( B, C, D \) form a Geometric Progression (GP).
3. We are given:
- \( A + C = 2 \) (Equation 1)
- \( B + D = 26 \) (Equation 2)

### Step 2: Express \( C \) and \( D \) in terms of \( A \) and \( B \)
From Equation 1, we can express \( C \) as:
\[ C = 2 - A \]

From Equation 2, we can express \( D \) as:
\[ D = 26 - B \]

### Step 3: Use the property of Arithmetic Progression
For \( A, B, C \) to be in AP, the middle term \( B \) must be the average of \( A \) and \( C \):
\[ B = \frac{A + C}{2} \]
Substituting \( C \) from Step 2:
\[ B = \frac{A + (2 - A)}{2} \]
\[ B = \frac{2}{2} = 1 \]

### Step 4: Substitute \( B \) back to find \( A \) and \( D \)
Now that we have \( B = 1 \), we can substitute this back into the equations for \( C \) and \( D \):
- From \( C = 2 - A \), we need to find \( A \) using the AP condition:
- Since \( B = 1 \), we have:
\[ A + C = 2 \implies A + (2 - A) = 2 \]
- This is consistent, so we can proceed to find \( D \):
\[ D = 26 - B = 26 - 1 = 25 \]

### Step 5: Find \( A \) using the GP condition
Now we need to check the GP condition for \( B, C, D \):
For \( B, C, D \) to be in GP, the ratio of the first two terms must equal the ratio of the last two terms:
\[ \frac{C}{B} = \frac{D}{C} \]
Substituting \( B = 1 \), \( C = 2 - A \), and \( D = 25 \):
\[ \frac{2 - A}{1} = \frac{25}{2 - A} \]
Cross-multiplying gives:
\[ (2 - A)^2 = 25 \]

### Step 6: Solve the quadratic equation
Expanding and rearranging:
\[ 4 - 4A + A^2 = 25 \]
\[ A^2 - 4A - 21 = 0 \]

### Step 7: Factor the quadratic equation
Now we can factor this equation:
\[ (A - 7)(A + 3) = 0 \]
Thus, \( A = 7 \) or \( A = -3 \).

### Step 8: Find corresponding values of \( C \) and \( D \)
1. If \( A = 7 \):
- \( C = 2 - 7 = -5 \)
- \( D = 25 \)
- The numbers are \( 7, 1, -5, 25 \).

2. If \( A = -3 \):
- \( C = 2 - (-3) = 5 \)
- \( D = 25 \)
- The numbers are \( -3, 1, 5, 25 \).

### Step 9: Calculate the sum of the first and fourth numbers
1. For \( A = 7 \):
- Sum \( = A + D = 7 + 25 = 32 \).

2. For \( A = -3 \):
- Sum \( = A + D = -3 + 25 = 22 \).

### Final Answer
The sum of the first and fourth numbers can be either \( 32 \) or \( 22 \) depending on the values of \( A \).