There are four numbers such that the first three of them form an arithmetic sequence and the last three form a geometric sequence. The sum of the first and third terms is 2 that of second and fourth is 26. What are these numbers?

Answer

To solve the problem, we need to find four numbers \( a, b, c, d \) such that:

1. The first three numbers \( a, b, c \) form an arithmetic sequence (AP).
2. The last three numbers \( b, c, d \) form a geometric sequence (GP).
3. The sum of the first and third terms is 2: \( a + c = 2 \).
4. The sum of the second and fourth terms is 26: \( b + d = 26 \).

### Step 1: Set up the equations

Since \( a, b, c \) are in AP, we can express the relationship as:
\[ 2b = a + c \quad \text{(Equation 1)} \]

For \( b, c, d \) being in GP, we have:
\[ c^2 = bd \quad \text{(Equation 2)} \]

From the conditions given, we can write:
\[ a + c = 2 \quad \text{(Equation 3)} \]
\[ b + d = 26 \quad \text{(Equation 4)} \]

### Step 2: Substitute Equation 3 into Equation 1

From Equation 3, we know:
\[ a + c = 2 \]
Substituting this into Equation 1 gives:
\[ 2b = 2 \implies b = 1 \]

### Step 3: Substitute \( b \) into Equation 4

Now that we have \( b = 1 \), we can substitute this into Equation 4:
\[ 1 + d = 26 \implies d = 26 - 1 = 25 \]

### Step 4: Substitute \( b \) into Equation 2

Now, we can substitute \( b = 1 \) and \( d = 25 \) into Equation 2:
\[ c^2 = bd \implies c^2 = 1 \cdot 25 = 25 \]
Taking the square root gives:
\[ c = \pm 5 \]

### Step 5: Find \( a \) using Equation 3

Now we can find \( a \) using Equation 3:
1. If \( c = 5 \):
\[ a + 5 = 2 \implies a = 2 - 5 = -3 \]

2. If \( c = -5 \):
\[ a - 5 = 2 \implies a = 2 + 5 = 7 \]

### Step 6: Summarize the values

We have two sets of values for \( (a, b, c, d) \):
1. When \( c = 5 \):
- \( a = -3 \)
- \( b = 1 \)
- \( c = 5 \)
- \( d = 25 \)

So the first sequence is \( (-3, 1, 5, 25) \).

2. When \( c = -5 \):
- \( a = 7 \)
- \( b = 1 \)
- \( c = -5 \)
- \( d = 25 \)

So the second sequence is \( (7, 1, -5, 25) \).

### Final Answer
The two possible sequences of numbers are:
1. \( (-3, 1, 5, 25) \)
2. \( (7, 1, -5, 25) \)