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------------------------------------------------ The sum of the first three terms of an arithmetic progression (AP) is 33. The product of the first and the third term exceeds the second term by 29. Find the AP.

a) Formulate equations representing the given conditions.
b) Solve the system of equations to find the common difference.
c) Verify the solution by calculating the terms of the AP.
d) Discuss the significance of the conditions on the AP.

To find the terms of the AP, we create two equations based on the sum and product of the terms and solve them. This gives us the first term and common difference, which we can use to verify the solution against the given conditions.

Explanation:

The question involves finding the terms of an arithmetic progression (AP) based on the given conditions. To solve the problem, we first need to establish a system of equations:

The sum of the first three terms is 33.
The product of the first and the third term exceeds the second term by 29.

We let the first term be a, the common difference be d, and thus, the second term is a + d, and the third term is a + 2d. Then we formulate the following equations:

a + (a + d) + (a + 2d) = 33
a * (a + 2d) = (a + d) + 29

Solving these equations simultaneously will give us the values of a and d. Finally, to verify our solution, we can substitute these values back into the original equations to ensure they satisfy the given conditions, which would demonstrate the significance of the conditions on the AP.