Let a, b and c be positive integers such that $\frac{b}{a}$ is an integer. If a, b and c are in geometric progression and the arithmetic mean of a, b, c is $b+2$, then the value of
$\frac{a^{2}+a-14}{a+1}$ is:

Question

Let a, b and c be positive integers such that $\frac{b}{a}$ is an integer. If a, b and c are in geometric progression and the arithmetic mean of a, b, c is $b+2$, then the value of 
$\frac{a^{2}+a-14}{a+1}$ is:

Accepted Answer

To solve the problem, we need to analyze the conditions given and verify why Option 2 is the correct answer.

### Step-by-step Explanation:

1. **Understanding the Geometric Progression:**
   - Let the terms of the geometric progression be $ a $, $ b $, and $ c $.
   - Since $ a $, $ b $, and $ c $ are in geometric progression, we have:
     $$
     b = ar \quad \text{and} \quad c = ar^2
     $$
   - Here, $ r $ is the common ratio of the geometric progression.

2. **Condition on Arithmetic Mean:**
   - The arithmetic mean of $ a $, $ b $, and $ c $ is given as $ b + 2 $:
     $$
     \frac{a + b + c}{3} = b + 2
     $$
   - Substituting the expressions for $ b $ and $ c $:
     $$
     \frac{a + ar + ar^2}{3} = ar + 2
     $$
   - Simplifying, we get:
     $$
     a(1 + r + r^2) = 3ar + 6
     $$
   - Rearranging terms:
     $$
     a + ar + ar^2 = 3ar + 6
     $$
     $$
     a + ar^2 = 2ar + 6
     $$
   - Further simplifying:
     $$
     a = 2ar + 6 - ar^2
     $$

3. **Condition on Integer Ratio:**
   - Given that $ \frac{b}{a} $ is an integer, $ r $ must be an integer.

4. **Solving for Specific Values:**
   - Assume $ r = 2 $ (since $ r $ must be an integer and $ a $, $ b $, $ c $ are positive integers):
     $$
     a = 2a \cdot 2 + 6 - a \cdot 2^2
     $$
     $$
     a = 4a + 6 - 4a
     $$
     $$
     a = 6
     $$

5. **Verification:**
   - With $ a = 6 $, $ b = ar = 12 $, and $ c = ar^2 = 24 $.
   - Check the arithmetic mean:
     $$
     \frac{6 + 12 + 24}{3} = \frac{42}{3} = 14
     $$
   - Given $ b + 2 = 12 + 2 = 14 $, the condition is satisfied.

6. **Calculate the Expression:**
   - Now, calculate $ \frac{a^2 + a - 14}{a+1} $ for $ a = 6 $:
     $$
     \frac{6^2 + 6 - 14}{6 + 1} = \frac{36 + 6 - 14}{7} = \frac{28}{7} = 4
     $$

### Conclusion:

- The value of $ \frac{a^2 + a - 14}{a+1} $ is indeed 4, which corresponds to Option 2.
- **Why Other Options are Incorrect:**
  - Option 1 (6), Option 3 (1), and Option 4 (3) do not satisfy the expression $ \frac{a^2 + a - 14}{a+1} $ when $ a = 6 $. The calculations clearly show that the only correct value is 4.

Thus, Option 2 is the correct answer, as it satisfies all the given conditions and calculations.