Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively, and the sum and the product of the elements of B be 36 and q respectively. Let d and D be the common differences of the A.P. in A and B respectively such that $D = d + 3, d > 0$. If $\frac{p+q}{p-q} = \frac{19}{5}$, then p - q is equal to:

Step 1: Represent the A.P.s and their sums/products

Let the three numbers in arithmetic progression (A.P.) for set A be $a-d$, $a$, and $a+d$, where $a$ is the middle term and $d$ is the common difference. Similarly, for set B, let the numbers be $b-D$, $b$, and $b+D$, where $b$ is the middle term and $D$ is the common difference. We are given that the sum of elements for both sets is 36. Using this, we can find the values of $a$ and $b$. We also express the products $p$ and $q$ in terms of $a, d$ and $b, D$ respectively.

Step 2: Calculate 'a' and 'b' and substitute into products

From the sum equations, we find that the middle term for both A.P.s is $12$. We substitute $a=12$ and $b=12$ into the expressions for $p$ and $q$. This simplifies the product equations, making them dependent only on the common differences $d$ and $D$.

Step 3: Use the given ratio to find a relationship between d and D

We are given the ratio $\frac{p+q}{p-q} = \frac{19}{5}$. We substitute the expressions for $p$ and $q$ into this ratio. After canceling out the common factor of 12 and simplifying the numerator and denominator, we cross-multiply and rearrange the terms to obtain a linear equation relating $d^2$ and $D^2$.

Step 4: Substitute $D = d + 3$ and solve for d

We are given that $D = d+3$. We substitute this into the equation $720 = 12D^2 - 7d^2$. Expanding and simplifying the equation leads to a quadratic equation in terms of $d$.

Step 5: Solve the quadratic equation for d

We solve the quadratic equation $5d^2 + 72d - 612 = 0$ using the quadratic formula $d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. We calculate the discriminant and find the two possible values for $d$.

Step 6: Determine the valid value of d and calculate D

The quadratic formula yields two values for $d$: $6$ and $-20.4$. Since the problem states that $d > 0$, we choose $d=6$. Then, we use the relation $D = d+3$ to find the value of $D$.

Step 7: Calculate p - q

Finally, we need to calculate $p-q$. We use the simplified expression for $p-q$ derived earlier, which is $12(D^2 - d^2)$. Substituting the values of $D=9$ and $d=6$ into this expression, we perform the final calculation.

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