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 p−qp+q=519, then p - q is equal to:
Get the complete, step-by-step math solution for: "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 su...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
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 p−qp+q=519. 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 d2 and D2.
Step 4: Substitute D=d+3 and solve for d
We are given that D=d+3. We substitute this into the equation 720=12D2−7d2. 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 5d2+72d−612=0 using the quadratic formula d=2a−b±b2−4ac. 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(D2−d2). Substituting the values of D=9 and d=6 into this expression, we perform the final calculation.