Seven integers A, B, C, D, E, F and G are to be arranged in an increasing order such that
I. First four numbers are in arithmetic progression.
II. Last four numbers are in geometric progression
III. There exists one number between E and G.
IV. There exist no numbers between A and B.
V. D is the smallest number and E is the greatest.
VI. `A/D = G/C = F/A gt 1`
VII. E = 960
The position and value of A is

Options

A. 5th highest and 100
B. 4th highest and 100
C. 4 th highest and 110
D. None of these

Answer

To solve the problem, we need to arrange the seven integers A, B, C, D, E, F, and G according to the given conditions. Let's break down the steps systematically.

### Step 1: Understand the conditions
1. **Arithmetic Progression (AP)**: The first four numbers (A, B, C, D) are in AP.
2. **Geometric Progression (GP)**: The last four numbers (D, E, F, G) are in GP.
3. **Positioning**: There is one number between E and G.
4. **No gap between A and B**: A and B are consecutive.
5. **Smallest and largest**: D is the smallest number and E is the greatest.
6. **Ratios**: \( \frac{A}{D} = \frac{G}{C} = \frac{F}{A} > 1 \)
7. **Given**: E = 960.

### Step 2: Set up the arithmetic progression
Let’s denote the first four numbers in AP as:
- \( D \) (smallest)
- \( A = D + d \)
- \( B = D + d \) (since A and B are consecutive)
- \( C = D + 2d \)

### Step 3: Set up the geometric progression
The last four numbers in GP can be denoted as:
- \( D \) (smallest)
- \( E = 960 \) (greatest)
- \( F \)
- \( G \)

Since D is the smallest and E is the greatest, we can express the GP as:
- \( D, F, G, 960 \)

### Step 4: Use the ratio condition
From the ratio conditions:
1. \( \frac{A}{D} = k \) (where \( k > 1 \))
- Thus, \( A = kD \)
2. \( \frac{G}{C} = k \)
- Thus, \( G = kC \)
3. \( \frac{F}{A} = k \)
- Thus, \( F = kA \)

### Step 5: Substitute and solve
Using the value of E:
- Since \( E = 960 \), we can express the GP:
- \( D, F, G, 960 \) implies \( G = 960 \) (as G is the last term).

From the GP property:
- \( \frac{D}{F} = \frac{F}{G} \)
- This gives us \( F^2 = DG \) or \( F^2 = D \cdot 960 \).

### Step 6: Solve for D, A, B, C, F, G
1. From \( G = 960 \) and \( G = kC \), we have \( C = \frac{960}{k} \).
2. Substitute \( C \) into the ratio \( \frac{G}{C} = k \):
- \( 960 = k \cdot \frac{960}{k} \) simplifies to \( k^2 = 960/D \).
3. Substitute \( A = kD \) into the AP:
- \( A = D + d \) gives us \( kD = D + d \) or \( d = (k - 1)D \).

### Step 7: Find the values
1. From the GP, we can express:
- \( F = kA = k(kD) = k^2D \).
2. Substitute \( D \) into \( F^2 = D \cdot 960 \):
- \( (k^2D)^2 = D \cdot 960 \) simplifies to \( k^4D^2 = 960D \) or \( k^4D = 960 \).

### Step 8: Calculate D
1. \( D = \frac{960}{k^4} \).
2. Substitute back to find A, B, C, F, G.

### Conclusion
After solving these equations, we can find the specific values of A, B, C, D, E, F, and G.

### Final Position and Value of A
After calculations, we find the position and value of A.