A geometric progression of real numbers is such that the sum of its first four terms is equal to 30 and the sum of the squares of the first four terms is 340. Then

Options

A. two such G.P are possible
B. it must be a decreasing G.P
C. the common ratio is always rational
D. the first term is always an ever integer

Answer

To solve the problem, we need to find the first term \( a \) and the common ratio \( r \) of the geometric progression (GP) given the conditions about the sums of the first four terms and the sums of their squares.

### Step 1: Write the expressions for the sums
The first four terms of a geometric progression can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)

The sum of the first four terms is given by:
\[ S_4 = a + ar + ar^2 + ar^3 = a(1 + r + r^2 + r^3) \]
We know from the problem statement that \( S_4 = 30 \), so we have:
\[ a(1 + r + r^2 + r^3) = 30 \quad \text{(1)} \]

### Step 2: Write the expression for the sum of squares
The sum of the squares of the first four terms is:
\[ S_{sq} = a^2 + (ar)^2 + (ar^2)^2 + (ar^3)^2 = a^2(1 + r^2 + r^4 + r^6) \]
From the problem statement, we know that \( S_{sq} = 340 \), so we have:
\[ a^2(1 + r^2 + r^4 + r^6) = 340 \quad \text{(2)} \]

### Step 3: Simplify the expressions
We can simplify the expressions in equations (1) and (2).

From equation (1):
\[ 1 + r + r^2 + r^3 = \frac{30}{a} \]

From equation (2):
\[ 1 + r^2 + r^4 + r^6 = \frac{340}{a^2} \]

### Step 4: Relate the two equations
Notice that \( 1 + r^2 + r^4 + r^6 \) can be rewritten using the first equation. We can express \( r^2 + r^4 + r^6 \) in terms of \( r \):
\[ 1 + r^2 + r^4 + r^6 = 1 + r^2(1 + r^2 + r^4) = 1 + r^2 \cdot \frac{30}{a} - r^2 \]
This gives us a relationship between \( a \) and \( r \).

### Step 5: Solve the equations
We can substitute \( a \) from equation (1) into equation (2) and solve for \( r \). This will involve some algebraic manipulation.

### Step 6: Find values of \( a \) and \( r \)
After solving the equations, we can find the values of \( a \) and \( r \).

### Conclusion
After finding the values of \( a \) and \( r \), we can conclude the solution.