Given the first three terms of the arithmetic progression (A.P) as $a^2$, $a(b - 3)$, and $4b$, we need to find the value of $\frac{b}{a}$.

Step 1: Understand the Conditions of an A.P.

For the terms to be in A.P., the difference between consecutive terms must be constant. Thus, the difference between the first two terms should be equal to the difference between the second and third terms:

$a(b - 3) - a^2 = 4b - a(b - 3)$

Step 2: Simplify the Equation

Expand and simplify both sides of the equation:

1. $a(b - 3) - a^2 = ab - 3a - a^2$
2. $4b - a(b - 3) = 4b - ab + 3a$

Now set the two expressions equal to each other:

$ab - 3a - a^2 = 4b - ab + 3a$

Step 3: Combine Like Terms

Combine like terms to solve for $b$ in terms of $a$:

$ab + ab - a^2 - 3a - 3a = 4b$ $2ab - a^2 - 6a = 4b$

Step 4: Isolate Terms and Solve for $b/a$

Now divide the entire equation by $a$ (assuming $a \neq 0$):

$2b - a - 6 = \frac{4b}{a}$

At this point, you can solve for $\frac{b}{a}$ in terms of the parameters or simplify the expression further given constraints $5 \leq b \leq 15$ and $a, b \in \mathbb{I}$.

Finally, solving this would yield the ratio $\frac{b}{a}$.

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Do you want further details on this equation or simplification? Here are some related questions for practice:

1. What is the general formula for the nth term of an A.P.?
2. How do you find the common difference in an arithmetic progression?
3. If the first three terms of an A.P. are $2, 5, 8$, what is the common difference?
4. How would the approach change if the terms were in geometric progression instead of arithmetic progression?
5. What happens if $a$ is zero in this context?

Tip: Always ensure to simplify equations properly and check for constraints when solving algebraic problems.